With the noted exception of yea, the other two words are very casual in scope. This almost lends itself to some casual errors, like using literally for emphasis (which is incorrect). Even when the written article is more casual, as would be the case with a blog post like this, it is still important to have good grammar, if only to ensure that you are conveying the meaning that you intend.
To make matters somewhat confusing, all three of these words really mean yes. However, they are used in different ways.
Yea (rhymes with day or pay) typically only comes up in the circumstances of a public vote. The common situation would be for someone to ask for all those in favor (of a particular proposition) to say, “Yea,” and all those opposed to say, “Nay.” It is very rare to see yea used outside of this “yea or nay” context.
Yay (rhymes with bay or dismay) is used primarily as an expression of enthusiasm, triumph, approval, joy, or encouragement. For example, you could say, “Yay! We won the game!” It could also be used as an expression of measure, as would be the case with “The box was about yay wide and yay tall.” Of course, this would only work with the corresponding hand motions.
Yeah (doesn’t really rhyme with anything) is an informal way of saying, “Yes.” It could be used in the context of having a lack of enthusiasm about a situation (“Yeah, I guess that would be alright”), or it could be simply a filler word to start a sentence or story (“Yeah, so I went to the store the other day”).
So yeah, only use yea when you’re voting on something. Yay! Now you can tell your friends to write the right word when they send out an update on Facebook or Twitter.