Is there an in-universe explanation as to why none of the playable characters in the Resident Evil games (except for those in Outbreak) become infected with the T-Virus? Seems like a massive oversight by the developers.
Rebecca Chambers is a playable character who is bitten but not infected.
The Resident Evil Wiki claims:
The chances of a human bitten by a t-Virus carrier (specifically a zombie) to be infected was 90% in the 1980s, though it is unknown if the Arklay Laboratory team (or any other branches in the virological bio-weapons project, for that matter) succeeded in increasing the chances of infection to Umbrella CEO Oswell E. Spencer's satisfaction. The lack of infection when Rebecca Chambers was bitten by an Eliminator could suggest that she was lucky enough to be naturally immune, or simply that t-Virus failed to enter her body when she was bitten.
Jill Valentine also became infected as a result of her fight with Nemesis, she eventually received a vaccine that allowed her to fully recover.
Jill faced Nemesis again in a battle outside the Clock Tower, where she was quickly infected with the t-Virus after the attack by Nemesis' tentacles...
...Jill regained consciousness and told Carlos that she has been infected by the virus. Carlos forced himself to search for Jill's cure or else she might become one of the undead. Jill also asked Carlos that if she turned into a zombie, he should not hesitate to kill her. After Carlos returned from a trip to the Raccoon General Hospital to find the vaccine, Jill was administered with the serum by Carlos and fully recovered.
There might be more. I'm not familiar with all of the games or characters.
To the question of why when playing if your character is injured why aren't they immediately infected.
While it may make for an interesting game mechanic to have your player get infected and die if ever injured, it sure would make the game hard. Since the game protagonists are expected to survive to the end, they logically won't become infected and die before that. - comments by Jack B. Nimble.
Resident Evil tends to follow a quite strict separation between the events that take place within the video cut-scenes (canon) and the events that take place during gameplay (not canon). This means that all of the things that happen while you're playing, including being attacked by zombies who're close enough to bite you, simply didn't happen, or, if they did happen, you were lucky enough not to be infected.
Note that when it's convenient (from a plotting perspective) for the main character to become infected with one of the various T-Virus variants, they're able to do so.
This one's slightly trickier. There's mention in one of the books seen in RE3 that herbs have an anti-viral component. That might account for your ability to stave off infection.
The bright green herb not only heals fatigue, but it also helps to strengthen the natural healing process so that wounds heal faster.
In addition, it has an active immune function that can be extremely useful against viral infection.
And there's also mention in RE5 of
where a zombie bites you but doesn't deliver an infectious dose. You may simply be getting lucky over and over again.