I can think of a few consequences of this type of system.
Missing half the conversation
A comment can't always be nicely categorized. A user could make a stupid pun and provide a valuable answer in the same comment. In this case, how do you classify the comment? Serious users could be annoyed that the pun is ruining their serious discussion. Punny users could be annoyed that their is a serious comment in their endless string of bad puns.
A serious user, in a severe case of the sillies, may be inspired to contribute their own bad pun and comment on the relevant information the user provided. Worse, a pun placing user, in a weird turn of events may suddenly be serious and ask a question about the information provided. Now you have the two groups interacting with one another in the same thread. Oh the humanity!
One such example of this type of discussion board is Slashdot. In a recent post, the top comment chain has posts that are (from parent down to child comments):
Insightful -> Funny -> Interesting -> Funny
By only looking at Funny comments, a user misses half of the conversation.
Slashdot comments are categorized by other users, not the author. In a recent post, the Insightful comments have gems like this:
Stop dissing America, it has the best democracy money can buy.
That isn't exactly the most insightful comment I've seen.
Forcing a hard split between serious/non-serious replies is going to create two sub-cultures on your platform. At some point, users will either get tired of context switching or not realize it's an option in the first place and end up stuck in one of the two modes until they either accidentally click out of it or someone points it out to them. At that point you'll either have users that only see silly puns or serious replies. That is going to create two very different impressions of your site.
If you browsed reddit and only saw the pun chains, I imagine you'd have a very different opinion of the site than if you only browsed it and saw serious responses.
On forums I've worked on and with, the job of the moderators has generally been to keep things civil, not to do work for the users. If moderators are expected to categorize a comment, you've greatly increased the amount of work you expect from them. No longer can a moderator scan through a post and look for potential problems, now they need to read (not scan) through every comment and determine if it's funny or serious.
If a comment is categorized wrong, what are the consequences to the user? Does intentional mis-categorization result in a penalty to the author? What about to the moderator?
On top of categorizing, the moderator still needs to deal with the separate subcultures that are developing. Do they deal with serious users differently than funny users? Does one group receive different responses due to how the moderator browses the site?
Slashdot has the type of system you're thinking of developing, with a few more categories. I recommend you really use the site for a while. Try browsing just one type of comment and see how the experience changes. Try switching between comment types and see what changes in the comments. Do you get more out of it? Then browse with all the comments shown and see, again, how the experience changes.