Question: 1. A particular mass of helium is required to lift a particular weight. If the weight stays the same and the mass of helium is decreased can the same lift force be achieved by increasing the acceleration of the helium gas? Not to be an Einstein, but if I recall correctly, force = mass \* acceleration.
2. Also, can helium gas be accelerated by heating it or by cutting it with a fan like propeller?
3. Lastly, when do we all get a flying saucer?

Answer: 
Normally when you speak of lifting by helium you are using the buoyancy of a balloon in the atmosphere. If you decrease the mass of helium you decrease the volume of the balloon. The lift capacity of a balloon is the difference between the weight of a balloon and the weight of the air it displaces. If you can lighten the envelope you can maintain that difference with a smaller balloon and maintain the lift capacity. The helium gas has no appreciable acceleration and there is no fan.

If you are doing something different to lift something with helium you will have to explain more clearly what it is.
Well, you could get more lift force out the helium by accelerating it. If the helium, let's say in a balloon, was initial moving upwards quickly and the object you're trying to lift was stationary, at some point the rope connecting them would go tight and at that point the rope will apply a downwards force on the helium slowing it and accelerating it downwards. The helium would apply an equal upwards force on the rope giving you extra lifting force.

You could test this experimentally fairly easily. Get a helium balloon, a string and a payload that is just a tiny bit too heavy for the balloon to lift. With the payload on the ground, pull the balloon down to ground level so that the string is slack then release the balloon. When the string goes tight the balloon should lift the payload off the ground but only very briefly. Once the balloon is done accelerating that force goes away.

The scenario described above is easier to picture with something denser and more streamlined moving upwards - say an arrow that's been shot up by a bow. The thing about helium is that it's got such a low density. This means that as soon as it's moving with any speed through air you are going to get substantial air resistance. Your rapidly moving mass of helium will end up exerting an upwards force on the surrounding air - not the desired result.