Why would the signal lights on a 1989 Dodge Aries not be working?

If the lights come on but do not flash then the flasher relay is defective, and needs replacing.

Check the easiest stuff first.

Check all your fuses at your fuse panel (AFAIK, it should be below the steering wheel, under the "Fuse access" panel.

Pop the panel off and check the fuses one by one to see if they are blown......Now, if you have missing fuses, that could be a problem....I know a lot of times people will "borrow" fuses for other accessories until they can purchase the right fuse, and will end up forgetting to replace the fuses :)

I found the following reference from a 1986 600 Owner's Manual.....it may or may not be the same fuses though! So use caution:

1: 20 amp yellow; hazard flashers

2: 20 amp yellow; back up lights, cornering lights, fan relay coil, auto load leveling

3: 30 amp C/B; power window motors, power convertable top

4: 30 amp green; A/C blower motor, heater blower

5: 20 amp yellow; park, tail, license light, side markers, display intensity, cavity 12

6: 20 amp yellow; stop lamps, dome light, map light, trunk/cargo compartment light, ignition switch key lamp time delay relay and light, power mirrors

7: 20 amp yellow; horn, horn relay, cigar lighter, glove box light, radio and clock memory, electronic chime

8: 30 amp C/B; power seat motors, power door lock motors

9: 10 amp red; radio and clock power

10: 20 amp yellow; turn signal lamps, heated rear window, console bin lamp, illuminated entry

11: 20 amp yellow; front windshield wipers and washers

12: 3 amp violet or 4 amp pink; cluster, A/C and heater control, heated rear window, ash receiver, console radio, message center illumination

13: 5 amp tan; fuel gauge, brake warning, engine warning, seat belt, voltage warning, message center, seatbelt chimes, speed control

14: 6 amp C/B; electric deck lid release, rear wipe/wash, liftgate rel.

15: empty

16: empty

And actually, when you yank that access panel off, shine a flash light under there, the fuses SHOULD be labeled.....BUT, they will not notate which Amp fuse goes in which slot, but you could cross reference with the slot # and accessories above :)

Once you've checked/verified all your fuses are OK......I'd go with purchasing new bulbs.....if the new bulbs don't work then you may have a connector problem and may need to change out the actual "socket" the bulb plugs into.