Sounds like the transistor driving this gate is stuck open. The gate is driven by a TIP122 transistor and these can sometimes go bad and cause a coil to remain energized. When you open the coin door, all high power is cut (for safety) and thus the gate closes. You'll probably see that the gate is even open when the game is in attract mode (with the coin door closed).
First, find out which transistor is driving that gate. The manual should have a table for this, it is a Q followed by a number. Find that transistor in the backbox on the big board.
You're gonna have to check that transistor with a DMM. Set your multimeter to diode testing (beep mode) and then ...
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From pinrepair.com:
For the most part, the most common testing will be on "darlington" transistors (which means they are actually two transistors in one package) such as the TIP102, TIP122, TIP36 transistors. Also most testing is done "in circuit" (installed in the circuit board). For each of these have one lead of the DMM on the metal case (tab) of the transistor (which is usually the center leg of the transistor). Then the other lead of the DMM will test the outside two legs individually. A value of .4 to .6, or 1.0 to 1.2 should be seen, depending on the exact transistor, and which lead (red of black) is on the metal tab. The biggest indicator of a bad transistor would be a value less than .2 (probably a short). Look to the specific pinball repair guides for more exacting details of this. Also remember, a transistor can ocassionally test as "good", but in fact be bad.
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You'll probably find it's shorted and you're gonna have to replace it. Don't stop there, but also check to see if the gate's coil is not burned. If it was stuck on for too long it may have melted inside and need replacing too.
Hope this helps!