Does this vulgar expression necessarily imply a certain body part?

Suck on it! usually implies fellatio.

But, it can be contextually clear you don't mean that.

If your thumb bothers you, suck on it.

It can also be used to say, take that:

I'm right and you're wrong. Suck on that!!

As in, keep that in your mouth and "enjoy" it for a while.

_{I decided to transfer what was meant as a comment into an answer because even constructive comments made without malice can be deleted because the author of a post can be offended by them. Political correctness runs both ways.}

It's normally "sucking one's thumb" or "thumb sucking" or "(s)he sucks her/his thumb", not "suck on it".

If the burn or cut is minor, you can suggest "try sucking on it". But in the OP's specific situation no one is going to think the speaker is referring to a thumb, a finger, or any suckable part of the human anatomy that is not the male genitalia.

The, primarily, American English expressions, suck it up, suck on it, are supposed to be vulgar.

It's possible that "suck on it" is related to "suck it up" which means to put up with something unpleasant. It is obscure as to its origins but there are suggestions from wiktionary and the Urban dictionary that it comes from: parade ground instructions to "suck in your stomach" when doing pressups; the expression "suck up your chest" meaning to take a deep breath and throw out your chest or from the necessity for a pilot to swallow his own vomit when he's been sick in his breathing mask.

Neither of these sources is exactly reliable and the idea that "suck on it" and "suck it up" are related is tentative but it is a possible meaning for the expression without the sexual inference.