I am running Curse of Strahd for my party and have used vampires very little in previous campaigns. Based on what I read for Strahd and vampires in the Monster Manual, I see many weaknesses laid out (sunlight, running water, etc). One thing that I don't see covered is holy symbols. Of course, in Curse of Strahd there is the Holy Symbol of Ravenkind which is an effective tool against vampires, but the question I have is for the more general idea of holy symbols (like those carried by clerics or paladins).  

In some general vampire canon, it is thought that vampires are turned away by holy items such as a Crucifix or Holy water. However, D&D is a world in which many deities exist. 

My question is are vampires affected in any way by basic holy symbols (like one carried by an adventurer or kept in a church) and if so is there a specific set of symbols that work against them?

simple rule: if it doesn't say it, it doesn't do it. Whilst, yes, in popular culture, the crucifix et. al. are effective against vampires, in DnD, they generally do not do so.

There are also examples in fiction of vampires ignoring a holy symbols, or ignoring them when not used by someone of faith. You can emphasize the use of a holy symbol in turn undead attempts against vampires, that certainly fits.

You can make the decision that vampires are affected by holy symbols without any mechanics involved. Rule that in your version of the universe, vampires experience noticeable discomfort around any sanctified symbol of a powerful deity, but this discomfort is not enough to affect their combat skills or attributes. This might result in some vampires targeting characters who carry such symbols and other vampires avoiding them slightly, but each time it is just a behavioural decision the DM makes on behalf of the creature, not an enforced mechanic.

You're running into rules-as-written, vs. rules-at-interpreted, vs. homebrew rules - which is pretty typical for a D&D question.

Everyone in this thread is correct here ( which is a minor miracle in and of itself! ) - it just all depends what your take is on the rules: there is nothing in the rules-as-written regarding holy symbols by themselves, although you can emphasize their use as part of the Turn Undead action, and you can homebrew a susceptibility to Holy Symbols for them any time.

The only thing I'd add, is that if you make Vampires susceptible to Holy Symbols, this is going to affect how challenging they are for the Party - so you might need to redesign encounters with Vampires in them, or give them a compensating homebrew ability to balance out their new homebrew weakness.