Is the universe finite?
Debatable. In mass, matter, and energy yes probably finite.
In time and space, infinite.
The Universe is expanding ( in fact is increasing in its rate of expansion) This means that the universe will continue to expand creating more of the space, time, matter and energy continuum.
The first law of thermodynamics is that matter and energy can not created or destroyed. This would mean that the amount of matter and energy the universe started with is the amount of matter and energy that the universe will end with. If the first law of thermodynamics is true then the amount of matter and energy is finite. fixed, though incredibly vast.
The second law of thermodynamics is that entropy will always increase in a closed system. If the universe is a closed system then the universe will continue to spread out, expand decreasing the order and complexity of the universe. Space and time are then infinite. ( however there will be no useable matter or energy at the end)