Do you have to sink the boat to complete Monkey Island 1?

In MI1, you can sink the boat where the three useless pirates are hanging out. I haven't seen this effect the gameplay in any way (I kept expecting to find them on the island somewhere). Then, in the end credits, I see what happened to these pirates, implying that I had to sink the boat in order to finish the game.

Where would I be stuck if I hadn't sunk the boat?

You can complete the game without sinking the boat. Should you not sink the boat, your crew will appear after LeChuck sets sail for Melee Island, and you will take your own boat back instead of Herman Toothrot's boat.

Canonically, you sink the boat. This is an important plot point early on in Escape From Monkey Island.

Where would I be stuck if I hadn't sunk the boat?

Sinking or not sinking the boat does not change gameplay in anyway. It simply changes some cutscenes, including the ending.

Sinking the boat

When leaving Monkey Island: You leave the island with Herman.

Ending: Meathook, Carla and Otis are shown captured by the cannibals.

Leaving the boat intact

When leaving Monkey Island: Meathook, Carla and Otis show up to take you back to Melee Island.

Ending: Herman is shown still stuck on Monkey Island.