We demonstrated a continuously tunable optofluidic distributed feedback (DFB) dye laser on a monolithic poly(dimethylsiloxane) (PDMS) elastomer chip. The optical feedback was provided by a phase-shifted higher order Bragg grating embedded in the liquid core of a single mode buried channel waveguide. We achieved nearly 60nm continuously tunable output by mechanically varying the grating period with two dye molecules Rhodamine 6G (Rh6G) and Rhodamine 101 (Rh101). Single-mode operation was obtained with <0.1nm linewidth. Because of the higher order grating, a single laser, when operated with different dye solutions, can provide tunable output covering from near UV to near IR spectral region. The low pump threshold (< 1uJ) makes it possible to use a single high energy pulsed laser to pump hundreds of such lasers on a chip. An integrated array of five DFB dye lasers with different lasing wavelengths was also demonstrated. Such laser arrays make it possible to build highly parallel optical sensors on a chip. The laser chip is fully compatible with PDMS based soft microfluidics.