If a ship flies on top of an asteroid and ends it turn on top of it I know you roll for damage, cannot do an action and cannot shoot. But then next turn it moves and it starts its turn overlapping the same asteroid, do you roll to see if you take damage again and are not able to do an action?

It depends. If you lay your movement template and it overlaps the asteroid then yes, you suffer the effects of overlapping again. If you lay your movement template and it does not overlap the asteroid then your do not suffer an overlap.

Your starting base position does not count for determining overlap. Only your movement template and your final base position.

Woe to the player who lands with his nose just over the edge of an asteroid.

Unless you have advanced sensors and a barrel roll to get clear before moving, you're going to lose another action and roll damage trying to get off it, yes.

Long story short: If you're going to hit an asteroid, make sure you've got enough speed to go THROUGH it.