Let's say I want to cover my ears. Is it 捂住耳朵, 遮住耳朵 or 盖住耳朵? And then if I want to say cover up a situation, is it 遮住事情 or 遮盖事情?
捂: cover small things almost entirely by a hand or hands
遮: cover large things partially or entirely by things like a sheet or a piece of paper
盖: cover something entirely by things like a lid or a cap
捂住耳朵 is better than 遮住耳朵 or 盖住耳朵 obviously.
The latter two are also acceptable and understandable.
cover up a situation: 掩盖/遮盖/遮掩/隐藏事实/事情, 掩盖 is the most frequently used word.
cover up a situation：掩盖事情 or 藏住事情 or 捂住事情
It depends on your situation.
If you don't want to hear something and cover your ears, it is 捂住耳朵.
If you don't want your ears from being seen and cover them, it is 遮住耳朵.
If you simply cover your ears, it is 蓋住耳朵.
遮住耳朵 and 蓋住耳朵 might be used interchangeably.
Both 遮住事情 and 遮盖事情 are right.
But if your intention is to hide something bad in deceptive way, the more precise word is 隠瞞事情.