First of all FiOS is NOT DSL. DSL is delivered over a copper wire pair with your POTS (Plain Old Telephone Service). FiOS delivers POTS and can delivery very high speed Internet, as well as Television over a fiber optic cable directly to your premises.
As to whether or not you can cancel your phone service and keep DSL it depends upon whether or not Verizon offers 'Dry Loop' service in your area. If they do, you can dump your phone service, although usually there is a discount on the DSL service for having Verizon phone service. In addition, if you have a Verizon contract for Phone and DSL, you are bound by the terms of the contract for however long the contract period is.
If you are talking about replacing DSL with FiOS, you can do that without changing your phone service. For example I have FiOS TV and Internet, but my telephone is still via copper wire with Verizon, and it is on a separate bill from the FiOS services. If my wireline telephone service were to be disconnected tomorrow, it would have no impact on my FiOS services at all. Generally if you upgrade from DSL to FiOS, Verizon waives any early termination charges on the DSL contract because you are upgrading to higher revenue Verizon service.
One other advantage to FiOS. It is NOT sensitive to your distance from the Central Office, nor is it sensitive to weather conditions. If your FiOS service works at all, you generally get whatever capacity you paid for, which can be as high as 50mb both ways....
Sparks82..jmw1950 is correct you can cancel phone service & it will have no impact on your internet service, but depending on your plan & how long its been that might change. You can always call customer service & find out.