Hello, I was checking my tire pressure today and when putting the cap back in, I accidentally dropped it into the front right tire and can’t get it out. Any advice? Thanks.
?? You mean behind the hub cap?
I assume you dropped it into the wheel and not the actual tire. If so just get another cap from AutoZone and don’t worry about it. The loose cap will ride forever or get tossed out.
Will it damage my brakes or anything? I’m not sure if it’s still in there.
NO
No, if it makes it that far it’ll just fall out.
Go for a ride around the block and when you get back it will be gone.
Make mine another vote to forgetaboutit. It cannot possibly do any harm. Unless it flies out onto a very weak and wounded ant. A pack of caps is dirt cheap, and they even sell them in Walmart in the bicycle area. They’re the same caps most American bike tires use. Now, if you have nitrogen in the tires yours might be green, but they’ll still be the same caps.
If you know anyone at all familiar with cars they can probably get it out.
Rotate your tires and it will turn up… 
If you have steel wheels and hubcaps, you just need to remove the hubcap to get it. If you have alloy wheels it is hust on the other side of the wheel and probably on the ground.
RUN AWAY!
RUN AWAY!
Tester
I used to put one of the metal caps with the stem unscrewer built into it (what IS the correct nomenclature for that anyway?), but then I realized that if someone wanted to bleed my tires I was providing the tool for them to do it. 
Congratulations on checking your tires. No problem, sleep well, but do buy a new cap to keep the dirt and ice out of the valve.
Good point, Bing. I second your congratulations.
...stem unscrewer...
Valve core tool.
Yup, that works!!
For the life of me I could not remember what it was called. I’m sure I’ll have forgotten by next week, but sincere thanks anyway. 
Thanks @the same mountainbike , now I can;t get the image of a poor little defenseless ant hobbling around on a tiny crutch and a bandage around his little head. At least he can use the valve cap to beg for donations.
Yosemite
LOL, now THAT is an image to imagine! :lol: