My 1986 BMW 325e five speed is stuck in gear. How

check shift linkage for damage or parts failure

Sticks shifts, that I love. for 50 year driving.
and restoring cars.
22 years old and can be repaired
not sure me if gear shift is direct to box or crazy cable drive.?
sure cable snapped.
if it is really in gear,
the clutch blown up, disk now just rivets no friction face there.
or PP pressure plate collapse.

if the MC/slave fails the clutch locks up sold.
yours is opposite ,it slips or shifter is bad (actual)
seem this box has remove ( to rear far) shift linkage
one must be sure this link works and not pull the box only too find out the link is dead, for simple reasons. bad link.
it looks like and old vw bug shifter, (1968 mine) but in reverse.
and vastly shorted link

see this photo.

arrow left pin broke or all parts on the right. bad,
all these parts go to the internal shift rails and forks.

Are you trying to test it for accuracy? If so, yes it's possible but you'll need special equipment to test it, plus you need to remove it from the vehicle (depending on year, it changed a lot from 1983 to 2013, add a year in the comments will be helpful).

Your simplest option is to drive some known distance and check to see if the odometer reads that you traveled the same distance, or drive beside a friend with a good speedo and see if it matches yours. Or, most GPS systems will calculate your speed too.

There are plenty of images in google as well.

E30 Fusebox diagram

http://www.e30tech.com/forum/showthread.php?t=54695

Most likely speedometer cable, check wiring harness to gauges and cable driven cruise control if so equiped

I would suggest you check the fuel filter or fuel supply to the engine as it may be a venting problem of a clogged fuel filter.
Cheers

Usually when a manual shift transmission starts jumping out of gear it's because the synchronizers are worn. The transmission will more than likely have to be repaired and or rebulit by a reputable transmission shop.

You can't use a tool to pull codes from an '87 BMW 325(e). You have the 2.7 liter and your DME, or computer is a Motronic. If you had a 325i, then a tool could read the codes because the computer was upgraded that year to a Motronic 1.1 system with a round plug in under the screw cap on the driver's side under the hood. The only tool you can buy is the one for resetting the service interval light and resetting the oil change light. To answer your question about going about finding your driveability problem would be to get a BMW Bentley Manual. This manual shows you how to use a standard voltmeter to check everything from your computer plug in. Your computer is located above the glovebox.

Good Luck,

~Stillin~