Here is a simple guide for using familiars with magic items that is consistent with the rules. Feel free to use some, all or none of what is here. If you are the DM its your world and you should feel free to do what is the most fun for your table. 

Can familiars use/attune to magic Items?

Yes they are creatures and can use magic items. They can also attune to them provided the meet any requirements the item has for attunement. 

How many?

Same as any other creature 3. There is no reason to arbitrarily handicap a creatures attunement slots just because it isn't a PC.

Adventures league games do have an added restriction: "...any item attuned to an NPC under your control (such as but not limited to, familiars, beast companions, simulacrums, conjured creatures, hirelings, lickspittles, etc.) counts against both the controlling character’s limit of three attuned items and the character’s permanent magic item count."   

This adventure's league rule isn't a bad guide to use in your home game if you like. And it will make your life easier as a DM. Basically the player (not the character) has attunement slots but any creature the player controls can use them up. So you and your familiar split your 3 attunement slots. It's fine and can save you some headaches down the road. But remember this is an Adventures League Specific rule for organized play. Home games don't have a limit on a character's permanent magic item count either. The reason they do this and why attunement slots exist is to limit the number of powerful items any one character controls. Having a character with access to 6 attunement items when everyone else only has 3 can shift the power balance in an unfair way. But you shouldn't be afraid to let familiars use and attune to magic items at your table. You just have to be more conscious of it

How Jeremy Crawford responded to the attunement question on twitter: "Do magic items attuned to a familiar count against the arcanist's total # of attuned items?"   J.C. - "The magic item attunement of your companions has no effect on the number of items you can attune to."

Whats a companion???? -  a person or animal with whom one spends a lot of time with whom one travels with. If you think in answering this question that Jeremy was only referring to a Rangers Animal Companion; he was not. Ask him yourself. He meant "companions" the word not the ranger ability. So this includes but isn't limited to familiars, beast companions, simulacrums, conjured creatures, hirelings, lickspittles, etc. (oh a lickspittle is someone who behaves obsequiously to those in power. Like a roadie or an entourage)

Does a familiar take a magic item with it when it disappears to its pocket dimension or when it dies?

Short answer: No. Long answer: Noooooooooooo. The ability for a familiar to be able to grab an item and disappear with it indefinitely can be game breaking. This allows for too many problems. Familiars can circumvent dungeons and traps or hide campaign important items. It creates a lot of issues if you let familiars do this. 

Now as a DM I would make the exception for items the familiar has attuned to it. Attunement creates a bond between the two. It makes sense that a an item bound to the familiar can disappear with it. Although It goes without saying that I wouldn't include Artifacts in this list as they are usually a special case for everything. This is my take as a DM at my table but feel free to make your own ruling. There is nothing wrong with a blanket statement of "No they can't take items with them ever."

If the familiar dies does it lose attunement to its magic items?

If it died then yes.... BUT.... Familiars don't die. This isn't clearly spelled out for you in the books, but when the familiar drops to 0 hit points, it disappears, leaving behind no physical form. That isn't the same as death though. If the familiar died the spell would say so. When players controlled character or creature drop to 0 hit points they start making death saves but they don't automatically die just for dropping to 0 hit points. Familiars at 0 hit points just disappear (presumably to its pocket dimension but the spell doesn't actually say where it disappears to). It reappears after you cast this spell again. A familiar is more likely to lose attunement to an item if it is more than 100 feet away for at least 24 hours (like being in a pocket dimension away from the item for 24 hrs)

Too Long Didn't Read:

Yes Familiars can use magic item. As for if they can Attune to the items, the easiest thing to Houserule is to not allow them to attune to their own magic items, but there is nothing against the rules by letting them attune to their own items. If you do, then you as a DM just need to be more intentional about watching the power balance between characters. Ultimately go by what is the most fun for you and your players. Work with your players to keep it fun for everyone. 

There is no reason to arbitrarily handicap a creatures attunement slots just because it isn't a PC.

Well, there kinda is and you state it here:

Having a character with access to 6 attunement items when everyone else only has 3 can shift the power balance in an unfair way.

Of course I get that you're not presenting these as hard and fast rules, but it seems a bit arbitrary to me that doubling the attuned items under a player's control is ok but sending any of those to the familiar's pocket dimension is game-breaking.

At any rate, I'm sure it's fine at many tables and ultimately a DM can control attunement by limiting the distribution of those types of items. But I'm pretty sure if I did it at my table, my players would abuse the heck out of it.

This is an easy fix. Familiars, companions, etc., can only have ONE attuned item. Your welcome :)

The problem there is that the more "companions" you manage to get the more attuned items you get. Like you befriend an army of commoners and have them attune to items.

In my game, the rule is that players can attune to a maximum of 3 items.

If you want your character's familiar to attune to one, that's fine, your character can now attune to a maximum of two.

Which magic items cause familiars to be so scary when attuned? I seriously want to see some of the ways that this could be used for or against the players? 

The ones on this list? Having PC controlled NPCs attune items is a problem mostly because it doubles your effective attunement slots as well as doubling the actions you have to use those items. It isn't necessarily the particular items; shenanigans could occur with almost any of the items.

A character I'm currently playing has traveled across a couple campaigns with the same DM.  Mostly because my schedule forced me to quit one, so he put me in another.  In the first one we rescued a Tressym and a Displacer Kitten.  The ranger got the kitten, my wizard got the Tressym.  But in that one, we were keeping her as a creature that was willingly being my familiar, not a conjured familiar.  So she could die.  My intent was to put the Amulet of Health he wears on her once I got his health/con score up.  Weather that be ASI or Feat accomplished.  So her wearing that amulet served me nothing, other than getting to keep my Tressym alive.  That I don't think should count against the PC's attunement slots.  Honestly, that's how I would use magic items on a familiar.  To make them more durable, not powerful.  I want them to be able to survive a fight so they can use the help action, Dragon's Breath, or carry items like potions to allies.  The point being, if the item isn't game breaking or balance shifting, then I don't think it should count against PC attunement slots.

DM let me make her a true familiar in the new campaign though.  So I can resummon her now.  Still, I'd put an Amulet of Health or such on her again.