There are technical, compatibility, legal, authenticity, and subjective reasons for not using it. I'm going to go through each in turn listing out the reasons with examples and references.

Starting with:

Hitler freaks out over Comic Sans

Technical Reasons

There are a handful of purely technical reasons not to use it.

The first, was a lack of italic variants. Comic Sans Pro attempted to correct this, and the new variants were merged into Comic Sans by Microsoft in Windows 8.

Second, Comic Sans just isn't a very good comic font, it has sub-optimal and wonky characters with poor kerning:

It should be noted that despite this, comic sans scores very high for readability in some studies - Diemand-Yauman, C.; Oppenheimer, D. M.; Vaughan, E. B. (2011). "Fortune favors the bold (and the italicized): Effects of disfluency on educational outcomes".

Comic Sans also comes in for criticism over reading disabilities, and as inappropriate for large blocks of text.

Here is architect small block on the left, and comic sans on the right: Now while Comic Sans is much beloved by many teachers and self-promoted readability experts, in fact you’ll see from the above passages that relative to the font to its left it has quite a number of fancy letter features, the other passage is more like what you imagine the diligent teacher would have written on the board. Probably, the belief you hear sometimes about Comic Sans being highly readable is really that the teachers, well, they just like it.

taken from http://typoface.blogspot.co.uk/2009/11/typefaces-for-disabilities.html#comicasb

Examples of fancy letter forms include the euro sign:

“The EU was going to sue us over that.” - Vincent Connare

While it comes in for criticism, it's also recommended for use by others. Few studies have been done to prove this however, and most support of Comic Sans justifies itself using simplicity, and personal anecdotes ( "well my brother is dyslexic and he says.." )

Of note, Comic Sans does have some features that help it in this regard, e.g. asymmetric glyphs for b/d, p/q etc

Here is the British Dyslexia Associations stance:

We asked dyslexia forum members. Only a few people responded. So it may not be a burning issue for most dyslexic people. It is likely that line length, line spacing and font size are just as important. Some loved Comic Sans, but others hated it. Some liked Century Gothic and teachers like purchasable Sassoon. On-screen and print preferences may differ.

More information on dyslexia and typefaces with regards to Comic Sans

Here is a Skeptics Stack Exchange question "Is the Comic Sans font easier to read for dyslexics?"

Microsofts own Comic Sans specimen at Comic Sans Café cites:

Comic Sans is the groovy script font which comes with the Windows 95 Plus! pack and is now available for the Apple Macintosh. Although it might be seen as a novelty typeface, which is great for titles, it's also extremely readable on-screen at small sizes, making it a useful text face.

The above quote and the full Microsoft Comic Sans website with full specimen can be found here

Compatibility

Compatibility-wise, Comic Sans is also not as widely available as people think. See this question on superuser for Linux support of Comic Sans MS, using comic sans does not guarantee compatibility.

If you're on Ubuntu, here is how you can install the Microsoft Core Fonts.

"If this page DOES NOT look like Comic Sans, you are probably using Linux!" - The Uncyclopedia page on Comic Sans

Legal

Legally, Comic Sans is not an open and free font, it's just very, very widespread, and there is a little known request for usage license from Microsoft & Monotype. Usage in css @font-face also requires a fonts.com subscription. Sadly the author does not receive royalties as Vincent was a staffer when he produced the font. The main reason people can use it is because Microsoft Apple and others pay for these licenses.

Authenticity & Sincerity

There are authenticity reasons for not using Comic Sans. While most of these are undone by the simple fact that Comic Sans is a digital font, the idea of comic fonts is that they are analogous to real-world attempts at lettering by hand.

E.g.

SMBC

Or this xkcd:

Now compare those to this comic set entirely in Comic Sans for an example of how effective Comic Sans is as a Comic font:

Some good points regarding sincerity were made as part of an Art installation called the Sincerity machine:

Click above to play

The sincerity machine was a typewriter set entirely in Comic Sans

Subjective

However, there are these subjective reasons not to use it:

Fonts with superior design and technical features such as Comic Neue exist

Comic sans was built for comics in an age of low resolution screens. It was never intended to be printed on billboards and placed in powerpoint slides

Comic sans is not a serious font, and use in serious applications is either satire or indicates a lack of seriousness or professionalism

There are more legible and readable fonts out there

Vincent Connare

There's also the testimony of the author of Comic Sans:

https://twitter.com/VincentConnare/status/8806817764

In his own words given to an article at deezen.com:

"I think people who don't like Comic Sans don't know anything about design," Connare told Dezeen. "They don't understand that in design you have a brief."

He later compares the font to pink tracksuits and Justin Bieber:

"There are 200-300 fonts installed on every computer but people pick Comic Sans because it is different and it looks more like handwriting and does not look like an old school text book," explained Connare. "It is a personal decision. The same could be asked of why do people like Ugg boots, Justin Bieber or pink tracksuits." "Comic Sans matched the brief, the brief of the entire Microsoft Consumer Division to put a 'Computer in Every Home' and to make something popular for the people of these homes and their kids. Comic Sans is loved by kids, mums and many dads. So it did its job very well. It matched the brief!"

For more information, here is a history of Comic Sans by Mashable