It appears Theodore didn't get along with his brother Constantine, with the latter being the preferred heir of John VIII despite being a decade younger, and eventually surrendered his rights to the throne in exchange for an appanage.
A son could dispute Constantine's succession (if born around the same time as his sister he'd be in his late teens/early 20s) and further weaken the Byzzies, he might even throw in his lot with the Ottomans and become their pet claimant. There's even a chance he could nab the throne himself, in which case it's hard to see him getting out of Constantinople.
Assuming he survives the fall of Constantinople he'd probably join his uncles in the Morea or go to Italy (which is where Thomas ended up anyway after Demetrius joined the Ottomans). In Italy he'd either attach himself to his mother's family (who seemed to be lords and condottieri on the Adriatic directly north of Rome), where it's not unlikely he'll end up dead in one of Italy's internecine wars, or he'll go to Rome and get a pension off the Pope (as did his uncle Thomas and Thomas' sons Andreas and Manuel).
Of course there's a chance he doesn't get along with Thomas/Andreas/Manuel and bickers with them over who's the rightful heir.
There's also the branch of the Palaiologos family still-reigning- as the Marquis of Montferrat- but the relation here is pretty remote (their closest shared ancestor is Andonikos II, who is Theodore's great-great-great grandfather). Nevertheless, he could hang out with them.
If Italy's not to his liking he could accompany his sister to Cyprus or another female relative to Russia, or he could even throw himself on the mercy of the Ottomans- his cousin Manuel (son of Thomas) did this in the 1470s and renounced his imperial rights in exchange for an estate and pension, whilst his uncle Demetrios received an appanage in Thrace after surrendering Morea. Any existence in the Ottoman Empire is rather fraught, though- a Sultan could decide he's a threat and summarily execute him on a whim (Demetrios managed to fall out of favour in 1467 and plunged into poverty).
So, he either goes to Italy to leech off the Pope/hang out with relatives, gives up and bows to the Ottomans, or goes to Russia/Cyprus/somewhere alongside a female relative who's getting married. He could pick up some minor lands here-and-there, but his descendants probably disappear into obscurity.