If one of the 4 rooms of the Starting Tile gets a 6 and explodes what happens? I suspect it's game over but I couldn't find any documentation about this one way or another. It is possible that the game just becomes extremely hard (until you POSSIBLY get the Dinghy placed) but there's no hinting that 1 room exploding causes the remaining 3 to, except inferance by the fact the Starting Tile contains 4 rooms that I see. I am also skeptical that if players are on those tiles they're just out of the game (seems against the spirit of the game or games of the "players keep playing genre").
For purposes of room explosions and consequences, each room is considered a separate "tile." If one room explodes, then that particular room on the starting tile is off limits, but the other 3 are accessible (until such time as they explode as well.) You can mark this by keeping the "6" on the tile, as the dice supply should be enough to have 1 die on each room in the game (should you be so lucky to have no explosions) You don't "flip over" the entire start tile, pirates on other rooms on the start tile aren't affected (pirates in the exploding room still suffer the normal consequences,) and the game doesn't end.
It seems that the OP is playing with the first edition copy of the game. The second edition does not have a "starting tile", instead it has 4 separate tiles for the starting rooms, allowing for each one to be individually flipped over, which clarified this rule. If you're playing the first edition, you can either keep the die on the room to indicate that it's impassable (or otherwise mark it) or you can visit this BGG file page, where the publisher has made a file of 4 "destroyed room" tiles you can use to cover them up.