Do unstable isotopes give off energy as they fall apart?
Yes, "falling apart" is radioactive decay. Radioactive decay is generally associated with energy emission.
One way of falling apart is beta decay. A simple representation of it is:
#overbrace(""_(0)^(1) n)^"neutron" -> overbrace(""_(-1)^(0) e)^"electron" + overbrace(""_(1)^(1) p)^"proton"#
The release of an electron from a neutron resulting in a proton is also associated with a small loss of mass known as mass defect:
# E = mc^2#
The loss of nuclear mass results in a release of (what was once binding) energy to the environment.
A common way of falling apart is nuclear fission. Nuclear fission is the process that powers the atomic bomb and nuclear power plants. This occurs spontaneously by obtaining a critical mass of the radioactive material
When an atom is bombarded with neutrons to split it into two smaller atoms, or when the atom spontaneously splits the smaller atoms have less mass than the "parent" atom (a mass defect).
So yes, when an atom "falls apart", the atom gives off energy.