Can I increase current using AC resonance?
If the passive resonant circuit has a high 'Q' then energy can build up in the circuit over many cycles. It will similarly die down naturally over many cycles. 

The intensity is not "ever increasing", however. As the intensity increases, so do the losses and at some point the losses equals the input power, so you have an equilibrium. 

Imagine a tuning fork. If you keep exciting it, the vibrations will build up (as will the losses) and at some point the metal shape might change if the excitation is powerful enough, but it's unlikely. Most real circuits have a rather lower Q than the mechanical Q of a tuning fork. 

In a real LC resonant circuit, the losses are usually due to inductor resistance, to core losses (if a core is used) and to electromagnetic radiation, especially at higher frequencies. Capacitor dielectric losses contribute too. Superconducting circuits can have huge Q's (in the thousands), resonant circuits made with parts from your favorite distributor, much more disappointing.