Most everything constantly mutates in the long run, it's just that viruses have such short life cycles (under optimal conditions, that is), reproduce in such extreme numbers and have such mutable genomes that they outmutate everything else. R-strategists in the extreme, they can more than afford to pay the potential losses brought on by deleterious mutations.
Mutation is one reason. Since viruses have a short generational time and produce millions of progeny they can easily create new strains or mutations that elude the immune system. Also, they are in a very hostile environment, so any change that increases fitness, i.e. by avoiding immune detection, will be immediately selected for.
There is one additional mechanism that make some viruses very hard to clear. Latency is when a virus becomes integrated into a host cell but does not express genes or make new infectious virus. This is the problem with HSV, EBV and HIV. Thus, some cells carry the genome, but avoid detection by the immune system because they are not actually making viral proteins. Only upon reactivation do they make new virions to infect other cells.
To control viral infections like HIV, cocktails are needed as this prevents the virus from adapting and becoming resistant. For HIV, 3 different inhibitors with different mechanisms of action are used. The virus will have to take a long time to figure out how to simultaneously become resistant to three different drugs. The same would be true for cancer cells and even bacterial infections. If one wants to treat superbugs in the future, I believe cocktails will be needed.
In terms of incurability, for retroviruses, others have aptly commented on why mutations aren't the reason the viruses are incurable. That is because they are integrated into the person's DNA. For other viruses, mutations and resistance to drugs makes them perhaps incurable in the short term.
High mutation rate, especially when it's associated to a high replication rate, is a cause of chronicity in some viral infections, such as hepatitis C or HIV infection. Nevertheless, it's not the only mechanism. Viral infection supression is highly associated to the expression of viral proteins on the cell surface, and the consequent action of citotoxic limphocytes. When a virus exists into a cell in a way that no proteins synthesis/expression happens (DNA integration in the case of HIV, episomes in the case of hepatitis B, similar mechanisms in the case of herperviridae), the immune system is not able to detect which cell is infected, and then the infection is not suppressed.
In order to persist, viruses will need a susceptible population so newborn become to be such population. Some viral genome mutations in some viruses certainly contribute for viral persistence but host susceptibility also contribute.