91 VOLVO 240 WILL NOT SHIFT INTO 4TH GEAR. WHEN I PRESS THE 3RD GEAR BUTTON ON THE SHIFTER, THERE IS A CLICKING IN THE DASHBOARD

U mean the shifter does not go in to drive? If that is so then replace the brake light switch on the brake paddle. Shifter is powered through the brake light switch.

Check vacuum lines for leaks.

This is common to the AW70 transmission when the rear bushing has been left very worn for a period of time. The rear shaft wear heavily on the main transmission housing causing the pressure not to build enough to engage the overdrive gear.

Does that transmission have external shift linkage? Sounds like a linkage broke or the neutral gate is way off.

4th gear disengagement:

The 4th gear is engaged automatically after the transmission has shifted through 1st, 2nd and 3rd gears

at certain speeds and loads. By pressing in the button at the rear of the selector level, the 4th gear can be

disengaged, thus providing a three-speed transmission. As a reminder the

? light on the dashboard

glows. By pressing in the button again, the transmission reverts to four-speed operation and the

?
light shuts off.
This is straight out of the owner's handbook. What this indicates is that there may be a problem with the shift solenoid. On the highway, does is seem to rev high? Can you press the button on the shifter and have the light go out? let me know.

the button on the top of the shifter is supposed to flip the" little lever"so that you can engage,leave it in N and you can drive it to get spare parts....

check gearbox oil level remove gear lever switch check feed to switch with meter or test lamp no feed check fuse if ok put 2 wires together car in 4th gear ignition on should hear click from overdrive solenoied if ok fit new switch if not check wires from lever to gearbox for breaks think you might find problem here if not check inhibitor switch and then solenoid hope this helps you

Check the transmission filter, may be clogged up, do not change the oil, keep the same oil when replacing filter, trust me to avoid transmission repair.

IPDusa.com.  If its similar to a 240 (and it most likely is) you can purchase a block off plate for the solenoid so you won't lose 4th gear (actually its OD from torque converter lockup).
http://www.ipdusa.com/Volvo-940960/Transmission-&-Drive-Train/Automatic-Transmission/Automatic-Over-Drive-Solenoid-Bypass-Kit/p-136-278-676-1031/

Just went thru this on my 93 240. There's a solenoid inside the glove box, on the left side that needed replacing, with age it started getting fussy, and one day I had overdrive, and the same afternoon I didn't, then next day it was fine, then it wasn't, and so on...Most annoying driving around in 3rd all day....Anyways, the part was about 75$ at a dealer, and installation was about 5 minutes...Solved the problem just fine...Also have your mechanic check the shift cable....it needed changing at the same time as the solenoid...jda/boston