Is it possible to build a breadboard device to simply view the code burned into a 28 pin or 40 pin DIP chip? I do not intend to do any erasing/programming on the chips, just wanted to view what is on them without the expense of a full-blown EPROM programmer device.
Jup, extreme simple, as EPROMs are static devices. Just connect a bunch of switches to the adress lines and some LED (use modern low power and don't forget the resistors) to the data lines, pull CS and OE and look whats on each address. You may even spare the switches and setup the address just with wires :)) I hope this isn't too low level.
Another way would be using that EPROM in some old computer. For a 2716 I'd pull out an Apple II and plug it into some extension card with an according socket (or instead of BASIC :)) and use the monitor to browse the content. YMMV.
While you can use toggle switches to enter addresses as @Raffzahn suggests (and it has a bit of the true retro authenticity to it: you'll be doing exactly the same operation that users of computers like the MITS Altair were doing to make them work), it's a lot of working.
The 74LS393 is a relatively cheap chip that contains a pair of 4-bit counters. They count on negative edge of a clock signal, so if you connect the high-order bit of one to the clock input of the other, it becomes an 8-bit counter: pulse the clock repeatedly and it'll count up to 255 then switch back to 0. Connect two of those together in the same fashion and you get a 16-bit counter. Hook this up to the address lines of your eprom and then you can just press a button to get the next value, rather than having to make a complex sequence of adjustments by hand every time.
Here's a page where somebody has implemented a similar device (in this case capturing the output of the EPROM into an Arduino so that it can be stored for later reprogramming) along with a diagram showing how to cascade the counters as I describe above.