I don't really like most God pantheons. They seem like overpowered superheroes and would like to develop a campaign with no Gods. Any ideas how to handle clerics, paladins and druids in this context? 

Paladins and Druids don't need gods. Paladins gain their powers from following their oaths, and druids deal with nature. A cleric on the other hand is screwed.

Have you read any Greek mythology? :D 

"Overpowered superheroes" isn't a bad description :)

Have you considered re-designing the deities - or the whole Theistic system in your world? It sounds like you have a feeling at least of how things should feel like; try making that.

I tried downplaying Clerics et al in a campaign world before - it didn't work particularly well for me or my Players.

For a system with clerics but no gods there's always the old Dark Sun setting.  Clerics there worship one of the four primal elements.

Like Arthur C Clarke's third law, most "gods" in fictional contexts tend to follow this general rule: any sufficiently powerful being is indistinguishable from a "god". Something or someone will need to take the place of a pantheon of gods, if only for the general observation that most civilisations operate with some sort of spirituality. As well, you may need a reason for the existence (or non-existence) of certain monster classes (celestials, demons, etc.), and it's really up to you how you want that to operate in your world. Some suggestions:

vague dualism - there is a "force" in the universe that can be manipulated for good or bad and the clerics roughly align philosophically. *cough*star wars*cough*
vague deistic monotheism - there is a silent creator who isn't particularly active at all, but aspects of this creator still respond either to faith or to 'spirituality' or to rituals to achieve certain outcomes. e.g. sacrifice 1 pig = higher chance of non-catastrophic farming weather, etc.
no gods but still religions - essentially, all the gods are "fake" or rather sincerely believed fictions, made up at some point in the history of a particular civilisation, but no one actually knows this, and yet, things seem to work, almost like it's just magic!

What kind of myths do the people in your world have? How was it created? If not gods, clerics can worship whatever it was that created things. 

They can worship ancestors. 

They can worship worldly forces (the sun, war, love) without there being a god attached to it. 

You can make there be one god, but there are disputes about the proper way to worship it, leading to as much or little  strife as you like. 

Quote from BobobSyxtieNein >>

Paladins and Druids don't need gods. Paladins gain their powers from following their oaths, and druids deal with nature. A cleric on the other hand is screwed.

 Clerics actually don't need gods either. Just like a paladin draws strength from their Oath, which is not necessarily to a God, a cleric draws power from their faith. That faith doesn't need to be in a god. It can be faith in a concept, ideal, or philosophy. 

Ravnica doesn't really have gods, and still has clerics and paladins. 

Quote from CharlesThePlant >> 

Clerics actually don't need gods either. Just like a paladin draws strength from their Oath, which is not necessarily to a God, a cleric draws power from their faith. That faith doesn't need to be in a god. It can be faith in a concept, ideal, or philosophy.  

You could take a page from Neil Gaiman's American Gods - where the idea is: when people worship, there is power. That power leads to the creation of a God. 

The book ( and mini-series ) is partly about the conflict between the Old Gods of nature, harvest, and fertility who came to the United States with the faith of the European immigrant ( the Norse/Germanic God Odin is prominent character in the book ), and the New Gods which have arisen out of North American culture like the gods of Television/Internet and Conspiracy Theories.

If you went down this route, then the Gods that are created by the worship of the people in your setting, are reflections of the cultures in that world. This still probably gives you Gods which look and act like "overpowered superheros" - but at least you have a rationale as to why the multiverse is structured that way. In fact this looks suspiciously like the Gods of Forgotten Realms. 

Or you could adapt that idea to the suggestions by Lunali and JCAUDM  , where you have the power-of-worship not manifest itself in a sentient deity, but in a general force which is philosophically/ethically biased in how it is expressed in the world. The Divine Domains could themselves become a "vague polytheism". It's possible that the ability of Clerics ( or any Divine caster ) to wield certain magic is dependent on the culture/worship of the people in the region,

From a role playing perspective, however - whatever you remove from the game, or alter, you should replace with something that has as much role-playing and dramatic potential for your Cleric/Paladin Characters. For example, If you remove sentient Gods in favor of elemental or philosophical forces, I'd play up the ceremonial and ritualistic aspects of the cultures & the Clerics ( I'm reminded here of Shinto which doesn't have "gods" per se' - but has a lot of ritual and ceremony ), to give your Players something to leverage in their role-playing.

I guess if you want to use words like “holy” and “divine” in a context without literal gods, then sure. But at that point is there really any practical difference?

If it walks like a god and quacks like a god...

That's a good question. Is the use of "the force" any different than D&D Arcane magic (apart from being more limited, but that's just the setting)?

Any person can swear an oath. What makes paladins special is that some sort of a holy presence grants divine abilities to act on that oath. It’s like saying a warlock doesn’t get powers from their patron, but from their pact. Maybe it’s true, depending on how you look at the nature of the relationship. 🤷🏼‍♂️

In one of the original editions of D&D (Red Box Mystara), there were no deities in the base rules, just one fundamentally Good-aligned system of churches: any actual religion behind this was glossed over. When I played, I loosely implied a medieval Catholic-like background with a single very distant god, but intentionally avoided any specific doctrines or deities. I think this might work really well for a lot of games, especially what it sounds like you're doing! Of course, the trappings of the churches could be different (e.g. Ancient Greek) depending on your setting, but the idea of an implied Good religion is the same.

I find that Players quite often "gloss over" the Deity of a Cleric, and tend to play Divine casters as Arcane ones, with a moderate layer of ethics laid over top.

I think having a few ( or single ) divine powers in the world ( good vs. evil ; dark side vs. light side ) with the real diversity, doctrine, and ritual being tied up in religious organizations to be an interesting one. The behavior of a Cleric might be under the scrutiny of their own Order, with their own secular consequences - which could provide some real interesting role playing "meat" for the DM and Player to dig into if the Cleric character is at odds with their own religious Order.  What does the Party do when the Inquisitor from the Order of the Shadowed Eye shows up because of magic wielded by the Cleric in a manner that contradicts Holy Doctrine? 

For some reason, that has a real World of Darkness  ( the White Wolf/Black Dog RPG setting ) feel to it, for me.

 I find it far more interesting when the cleric is not part of the religious order.  The order is effective (or not) at dealing with the day to day issues that people encounter, but when god needs a champion, they pick someone else.

I like that as well - and it definitely plays up the "characters as heroes" trope. I think it boils down to the tone and feel you want the setting to have. Like I said, I think the "oppressive religious order" model feels much darker - hence the reminder of the World of Darkness setting.

I'd be happy running campaigns under both models; I think they both add their own flair.

Perhaps you could do both, based on tier. Low level Clerics are subject to the constrictions of their Order - but once they obtain a level of experience and power, they are inducted into "higher orders" within the Church which act as direct Divine Agents - not subject to any restrictions except perhaps the highest levels of the Church? Just spit balling ideas here :) 

Quote from Vedexent >>

Perhaps you could do both, based on tier. Low level Clerics are subject to the constrictions of their Order - but once they obtain a level of experience and power, they are inducted into "higher orders" within the Church which act as direct Divine Agents - not subject to any restrictions except perhaps the highest levels of the Church? Just spit balling ideas here :) 

 If you want the PCs associated with the church, that sounds like an interesting way to go.  If you'd prefer the church as a neutral/opposition party, NPC casters within the church could actually have bard or paladin based spellcasting while a PC not affiliated becomes a cleric because the god believes in them instead of in the church.