To be more specific: Suppose a space station is orbiting in a cislunar orbit (about 1000 miles away from the Moon). Now you keep adding more mass (huge cargo) to the cislunar space station. The space station departs for Mars with that huge cargo (much more mass). There will be fuel needed for putting the ship on the trajectory to Mars and initial change of velocity. But will the fuel required be much higher or the same if the mass is added, say, 1000 times more? Thank you for asking for clarification.
In short, no.
Velocity implies direction. If you're in a circular orbit of a body (planet, star, etc), then you will maintain the same speed, but direction will constantly be changing as the body's gravitational pull curves your path.
In some cases you want to fight gravity, for instance hovering above an asteroid. In this case, since you're fighting gravity, this would require use of fuel.
Although they made some additional comments, the first 2 answers assume orbital travel - which is of course what we usually think of when we talk about microgravity.
Microgravity is the condition we experience when we are under the influence of a constant gravitational field (possibly the earth's) as well as the influence of other minor fields and forces. Examples of these others are the distribution of the mass of the spacecraft we're in, and the friction of any residual atmosphere. In the case of the "vomit comet" they include the fact that the plane's trajectory never cancels the earth's field exactly.
As per Newton's laws of gravitation and motion, under the influence of any gravitational field, micro or not, velocity cannot stay the same; it must change. In a perfect circular orbit, the speed will stay the same, but the direction will change.
In other words, if there is any gravity, micro or not, you always need fuel to keep the same velocity. In a circular orbit your speed stays the same, but your velocity does not. In remote space, far from any stars, your velocity may be nearly constant, but it will still change slowly. Hence you still need a force, and thus fuel, to keep it constant.