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Thrust is independent of weight. In straight and level flight thrust operates forward, drag operates to the rear, weight operates downwards and lift operates upward. For a rocket, traveling straight up (away from the center of the earth), drag and weight operate downwards while thrust operates upwards.

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The above answer appears correct (where "operates" is taken to mean "exerts force in the indicated direction") - additional discussion follows:

(I am assuming we are talking about a rocket or an aircraft for the purposes of this answer. I am also assuming that by weight, we mean mass)

The mass of a craft does not affect the thrust per se, that is, they are independent quantities as discussed below.

Since the thrust is a function of the type (efficiency) of engine and the amount of fuel delivered per second, a formula expressing the thrust will not include the mass of the craft directly, hence the independence.

However, of course, the thrust that a given craft can generate will be a function of the size of its engines, the type of fuel, and the amount of fuel delivered to the engine per unit time. Since the weight of the craft is partly determined by the size of the engine and the specific gravity of the fuel, there is a second order relationship between the weight of the craft and the thrust.

The acceleration of the craft under thrust will of course depend on the thrust and the mass according to Newton's Second Law, F=MA, or in this case, the form A=F/M.

PS: I'm not considering the relativistic realm either - you're on your own there!

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