There are countless possible mechanisms: mutation of the kinase so the inhibitor doesn't bind anymore, over expression of the kinase targeted, over activation of the inhibited pathway, activation of the pathway independent of the inhibited kinase, shift to isoforms that are weakly or not at all targeted by the drug, shift of the cells so the inhibited pathway is not essential for cell growth anymore, expression of a drug pump to clear the drug, expression of pathway that katabolizes the drug, ... And probably some more that I forgot to think of right now.
Do you have a special inhibitor in mind or are you just interested in general resistance mechanisms? These may include, but are not limited to: up regulation of bypass-pathways, increase of target kinase expression, point mutations of the target kinase affecting inhibitor binding, upregulation of inhibitor degradation, reduced inhibitor uptake into the cell, mutations of the kinase affecting enzyme velocity or effective substrate binding......
The resistance to kinase inhibitors are due to selection of possible kinase mutants, which are generated during cancer progression and then the inhibitor do not bind the kinase anymore. By time the mutated kinase increases in the population. The cancer progress even faster. These mutants can possibly targeted by newer drug generations.
There are countless possible mechanisms: mutation of the kinase so the inhibitor doesn't bind anymore, over expression of the kinase targeted, over activation of the inhibited pathway, activation of the pathway independent of the inhibited kinase, shift to isoforms that are weakly or not at all targeted by the drug, shift of the cells so the inhibited pathway is not essential for cell growth anymore, expression of a drug pump to clear the drug, expression of pathway that katabolizes the drug, ... And probably some more that I forgot to think of right now.
Rightly said by all the previous answers.... the question is very general and has a lot of possible answers. You will find many references to all the possible mechanisms of resistance to kinase inhibitors... though each mechanism can be valid for all the drugs, each drug has its own set of common mechanism of resistance.
You could find clinically relevant and disease specific mechanisms on the following site: http://mycancergenome.org/
One of the underlying resistance mechanism is over activation of the kinase upon dissociation of the inhibitor. This may be overcome by targeting allosteric sites rather than the ATP binding site
If you are using a kinase inhibitor then you could attribute it to mutations, additional requirement for the pathway to be inhibited not met (for instance if dual kinases are invovlved inhibiting one of them is not going to be effective), downstream effector not being depepndent on the kinase of interest alone and being turned on by other crosstalk pathways or an alternative pathway that gets activated subsequent to inhibition of your intended pathway. Hence the effect that you may want to achieve may never been see in the cell based system.
For a topical example of kinase inhibitor resistance in cancer, you might look into the mechanisms of acquired resistance to mutant BRAF V600 kinase inhibitors in melanoma.