In Journey to the Center Of The TARDIS, the TARDIS crashes in the ship of Gregor, Bram and Tricky. After the Doctor convinces the three to enter the TARDIS, they don gas masks and blast suits. The Doctor also puts on a gas mask, but omits the blast suit. My friend told me it was because "He's the Doctor," but I never took this to heart. Why didn't the Doctor put on a blast suit? Was it somehow because of his Time Lord biology?
Time Lords in general are highly resistant to things normal humans are, like for instance radiation.
An example of this is different forms of radiation such as Rontgen in the episode Smith and Jones he talks about how he used to play with bricks of them in the nursery.
Not to mention that overall Gallifreyans in general are very resilient to a number of different things like the cold as seen in the episode The Tomb of the Cybermen.
Also when they went into the Tardis there was just a lot of smoke and gas that was leaking so it was funneled out by The Doctor using fans. Further into the episode the only real danger that is posed is from the Eye of Harmony which creates the creatures seen in the episode explained by the Doctor when he states that The TARDIS was trying to save it's passengers and inadvertently causing their cells to burn creating the said creatures.
One last thing to mention is that we don't know how much the blast suits protect their users anyway. The humans had no idea what was waiting inside of them in The TARDIS so they went as prepared as they thought they would need. The Doctor himself knew better, if the amount of radiation or dangers were lethal even to him he would have worn some sort of suit as well. And given that The TARDIS was in fact rather somewhat intact on the inside besides the gas and fumes leak it can be seen that there wasn't any more form of danger as far as radiation or the like.