Any character with a healing factor presumably also requires superhuman pain tolerance, to avoid falling unconscious from the crippling pain and then waking up ten minutes later good as new. Still would be useful, but not so much in a combat situation (where being unconscious makes you useless at best, and makes you a liability or opens you up to a Coup De Grace at worst.) It's not really necessary to make the power work, though, so not actually a required secondary power, and there are many examples of healing characters who don't have abnormal pain resistance.
More importantly, they'd burn up a lot of energy accelerating the healing process that fast, and so would need super-stamina and more efficient internal energy use than normal (or an alternate energy source) or their own power could kill them by draining their body's resources too fast. And to regenerate takes mass, so either they have limits to what they can regrow at one time, or access to some other source of mass, like Another Dimension—see Shapeshifter Baggage.
Also, they'd need a way to fight infection and other foreign substances. Having your guts spilled all over the floor would open the door to all sorts of germs and viruses, not to mention dust and such. There's a reason hospitals are super sterile. Conversely, they get infected like anyone else, but recover instantly due to the healing factor producing antibodies at an accelerated rate. Wolverine's healing factor was used to develop antibodies to a deadly virus on at least one occasion. Most likely they would also have some kind of super-genome, considering that so much rapid healing and DNA replication would increase the possibility for cancer-causing genetic defects massively. This would explain Deadpool's Massive scarring, since his skin is both spreading the cancer and healing the damage caused by the cancer And, cancer aside, certain kinds of attacks could reasonably be assumed to cause genetic damage to the tissues which the regenerating tissue is growing from.
In addition to that, it is possible that such a healing factor would have to display some sort of Morphic Resonance (or at least be controllable). Having broken bones heal almost instantly sounds much less impressive when you realize that this gives almost no time to set the bone before it heals crooked. On top of that, if you got shot you would have to take the bullet out yourself, then wait for it to re-heal. Ouch.
Let's not forget when things that are wrong with you body heal as they were. Deadpool gives us a pretty bad look at what happens when your healing factor heals cancerous cells or brain abnormalities as if they were normal healthy cells.
- Technically, it's not always necessary to remove a bullet, but serious superheroes would probably end up getting lead poisoning if they didn't do a spring clean every now and again