Teleportation via teleportitis works like other sources of teleportation, being blocked by no-teleport levels and sometimes by the Amulet of Yendor, controlled by teleport control, and landing you somewhere "safe."
Teleportitis is significantly more useful with teleport control, from a ring or an intrinsic. "Controlled" or "uncontrolled" teleportitis refers to whether a character has teleport control along with his teleportitis.
It may be gained by eating some corpses:
- leprechaun 50%
- n nymphs:
- tengu 20%
- Wizard of Yendor 25%
Or wearing a ring of teleportation (these are often cursed, so be careful). Or eating a ring of teleportation.
Note that the Quantum mechanic doesn't give teleportitis (but toggles the speed intrinsic instead).
Every game turn, there is a 1/85 chance you will try to teleport. This does not cost nutrition or energy. Unlike most other forms of teleportation, this happens at the beginning of your turn, rather than at its end, which means even very fast monsters will not have time to react to your approach. 
Teleporting at will
If you have teleportitis and you have reached experience level 12 (or level 8 for a Wizard) then you can use ctrl + T to teleport at will (Note: this does not grant teleport control). This costs nutrition and energy like a spell; in 3.6.2 the energy cost is the same as for the spell (30 Pw), while in 3.6.1 and earlier it only uses 19 Pw. It costs 100 nutrition (unaffected by hungerless casting), except in some rare cases where your pet prevents you from teleporting (see below). These costs are still incurred if you choose not to teleport by declining to pick a location (in the case of controlled teleportitis), or even if you cannot teleport because you are on a non-teleport level.
This can be useful; for example, with controlled teleportitis, you can repeatedly teleport to rapidly decrease your nutrition, so as to eat a set of giant corpses before they rot away.
If your leashed pet is standing too far away from you, cannot be moved next to you, and the leash is cursed, you will be prevented from at-will teleport with no nutrition cost. The same will happen if you are riding and your mount carries the Amulet of Yendor. Both cases will still cost Pw however.
If you cannot teleport at will, then ctrl + T will attempt to use your teleport away spell instead, or will make use of a teleport trap if you happen to be standing on one. (Source code: teleport.c#dotele)
Another cure is to eat disenchanter corpses. Each time you eat one, you have 1/11 chance of curing teleportitis, otherwise you may lose another intrinsic. Just like with gremlins, you will likely lose a lot of other useful intrinsics before you get rid of this one.
Shopping with uncontrolled teleportitis
- Stand on the item you want to buy and use the #chat command; the shopkeeper will tell you the price (eg, "a food ration, price 60 zorkmids.")
- Drop that much gold to the shopkeeper by using a numerical amount with the drop command (eg, "d60$"). The shopkeeper will recognize it and establish a line of credit for you (eg, "You drop 60 gold pieces. 60 zorkmids are added to your credit.")
- Try not to throw gold at shopkeepers. There's a chance that the shopkeeper will get angry with you.
- Pick up the object. If you are randomly teleported out of the shop (or leave it in any other way) at this point, your entire credit will be used up to pay for the item without angering the shopkeeper (eg, "Your credit of 60 zorkmids is used to cover your shopping bill.") Note that if your credit is greater than the amount owed, the shopkeeper will keep the difference.
- Alternatively, just use the pay command as usual, and the shopkeeper will use up your credit before asking you for any more gold (eg, "The price is deducted from your credit.")
Teleportitis can be acquired by consuming some other corpses (not an exhaustive list):
- s phase spider 40%
- n brownie 80%
- n Aphrodite 100%
- l leprechaun wizard ??%
- P Giant shoggoth 20%
- e Blinking eye 80%
- @ Gypsy 100%
- Q doctor Frankenstein 100%