I'm voting 'Yes', for 2 main reasons.
(I'll mainly be focusing on likes here, just because I've seen more problems arise out of likes than messages. Some of the points I make, however, may still apply to messages.)
1. TCF is
NOT Terraria Online.
Terraria Online had it's own fair share of problems with competition, though it was even more concentrated around likes than messages (Forum Games were super OP, so no-one really took the messages tab
too seriously). There were a lot of the problems around 'like farming' in TO that weren't addressed, mainly because no-one saw any harm in it at the time. But it did arise to a lot a competition between members with massive like counts, with one member even trying to obtain the unobtainable 'Like goal' trophy.
Is that healthy for a forum? In my opinion, no, it is not.
As already mentioned, this was mirrored here in TCF for a time, until moderator intervention slowed that to a halt. But why did it start in the first place? *Hint hint nudge nudge*
2. What is the point of a like?
Well, as Sigma said (to take him a little out of context):
Likes can mean "I agree", "Thank you", "That's funny", "That worked", "That's helpful"...
They're not there to see who's better. They're not there to reach a certain 'like goal'.
They're there to say something without actually saying it.
That's my two cents on the matter. Now, on to actually voting in the poll...