What happens to the invariant quantities in special relativity?

It depends on what these invariant quantities are. It is not enough to specify that something is invariant, you also need to specify under which operation these quantities do not change (= are invariant).

In special relativity there is an operation called a Lorentz transformation which applies the effects of a speed increase, thus applying time dilatation and length contraction. A Lorentz invariant quantity is a quantity which remains the same under this transformation, i.e. it has the same value for every observer in an inertial frame.

Examples of such invariants are the lengths of four-vectors, the generalizations of the common 3-dimensional vectors such as those indicating place and momentum. For example the 3d-vector for location (x,y,z) is joined with another quantity for the time dimension into a 4-vector whose length is Lorentz invariant. There are more Lorentz invariant quantities, some of them quite complex.