Most of the replies seem to focus on the merit of the endeavor rather than directly answering the question, so I'll go out on a limb and say that it's possible that you could make the swap. To make that determination , you'd want to look at the existing vs replacement chip pinouts as shown in the MGF's(eg. TI or AD etc) catalog cut sheets. You may find that families of DAC's share common architecture(even between MGF's), and so allow for direct substitution. Even if there are differences in pin locations but identical functionality, you can still do it by piggybacking the new chip on top of the existing one. Note that some pins may be unneeded for your application.
To do this, you cut the pin(s) to break the connection(s) that supply power to the existing chip, then, use fine gauge wire(ie wire wrap wire) for jumpers. Verify that there are no conflicting polarities between the old and the new chip's pinouts for the remaining uncut pins. Alternately, you can bend up the unmatched pins of the replacement chip and still use jumpers; if you wanted to remove the existing chip altogether. There should be no need however, since the input impedance of the existing chip and those associated should be quite high, allowing for the additional fanout. Using the piggyback technique saves you from having to rework(reflow) the board. You can use a conventional soldering iron using the piggyback method.
If you do choose to rework the board, it's still very possible using an oven, paste and desoldering equipment, which can be obtained rather inexpensively on ebay, Jameco, Digikey etc. Rework kits and equipment are available as well. The rework comes down to heating the existing chip to remove it, then applying solder paste to the pins of the new chip and setting it in place over the pads on the board, then putting the whole thing in a ~400F oven(eg toaster oven) or use a heat gun. Alternately, you can cut the pins off the existing chip to remove it, then use a conventional soldering iron(or hot air gun) to remove the pins individually with tweezers. This latter approach is my preferred method since it doesn't involve repeated heat cycling of the entire board.
One other thought is that you could solder leads directly to the existing DAC chip pins to bring the digital signals out to a header. From there you could build a piggy back board or externally convert it to a serial bitstream/USB output etc. as required for use with a conventional standalone DAC.
I've had good success repairing and modifying SMD boards using these techniques. It's not hard or particularly expensive, but requires some patience and maybe magnifying glasses(or excellent vision). As others have said, unless your existing DAC is truly awful, you'll likely not hear the difference, but if you want to do it for fun, it's very possibly done. The methodology of signal conversion interface is pretty well established and fairly standardized, so there is usually more commonality than not between devices.