First let's say that all velocities are relative to the centroid of mass of the Universe. Given that every body in the universe is accelerating apart from every other, and mass increases with velocity as defined by special relativity. As the speed of every mass in the universe approaches the speed of light, then as I understand it, the masses increase to near infinity. At some point, would the gravitational field of every body eventually pull everything back to the center of the universe? Or, are the masses of the universe all expanding apart with a uniformity such that the average velocity of every mass relative to the center of mass of the universe is zero, and the
There is no centre of mass for the universe because it is isotropic and homogeneous. That is every point in the universe is like every other point so there is no one point you can point to and say this is the centre. For more on this see Did the Big Bang happen at a point?.
However we can certainly pick any point in the universe as the centre for our coordinate system then measure the velocities relative to this point, and we do indeed find that the velocities of distant galaxies are related to the distance from our origin by the Hubble formula:
$$ v = H_0 d $$
And there is indeed a distance at which $v$ becomes equal to the speed of light. This distance is called the Hubble horizon.
However the mass of a moving body does not increase. This was an old idea based on the fact that the relativistic momentum is not proportional to the velocity. Fast moving objects do not increase in mass and do not have an increased gravitational field. So the argument that the universe must recollapse due to an increase in mass of objects near the Hubble horizon is a fallacy.
Universe is expanding because the space itself is expanding. The expansion is not due to acceleration of the bodies with respect to space. Therefore, the bodies are getting away from others with the expanding space, not due to their own acceleration. So, their speed with respect to space is not changing and so, the mass approaching infinity does not apply. In fact, if we look far enough, the expansion may already have crossed speed of light, but that is expansion of space over that much distance, not the speeds of the bodies with respect to space.
Speeds faster than c, wrt space, are not possible per relativity.
Although everything falling back together in the future is not ruled out per some theories predicting big crunch. Even if that is to happen, it will not be due to the relativistic mass.