I just paid off my car - should I release my Credit Union's lien on the car?

For practical purposes, you will need/want to remove the lien before you sell the car¹, so unless you plan to run the car until it falls apart², you will need to remove it at some point. The question then becomes: do I remove it now, or wait until later?

I've not been able to find any reason why keeping the lien attached to the car's title would be beneficial: Experian's article What to Do Once You Pay Off Your Car starts:

Get Your Car Title

You just paid off your car and own it outright—now get the paperwork that says so. Your car title is a piece of paper that lists the official owner and any lien holders on your car.

before going on to discuss the differences between title-holding states and non-title-holding states. In either case it continues:

Once you receive either of these documents, follow your state's protocol for transferring the title to your name. This will allow you to show ownership and sell the car in the future, so get all this paperwork in order as soon as possible.

(It also notes that you may get a better rate on your insurance, backing up Kwadzo's answer).

What it does not give is any reason for not removing the lien, or for waiting to remove it, and neither does any other site I've found.

With that in mind, the only half-plausible reason I can think of to not remove the lien straightaway is to "not have to fork out $20" (at least for the time being). However, although I've no experience with any of America's Departments of Motor Vehicles, the experience I do have has taught me that such fees almost never get reduced. If you leave the lien in place "to save $20 today", then by the time you come to sell in a few years time, it may very well cost $24 to remove the lien from the title.

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¹ According to Credit Karma's article How to sell a car with a lien, it's possible to sell with a lien attached, but is much more hassle.

² According to Loren Pechtel's comment, you still need "clear title" if you want to junk an old car. This is generally confirmed by a Junk Car Medics article 5 Things to Do Before You Junk Your Car for Cash to an Auto Junk Yard:

2. Get Your Title In Order

You need to return license plates and cancel your insurance. The single most important thing you need in order to transfer ownership of your vehicle to the junkyard is a valid title to the car. You can sell your car without a title but it’s much harder.

While acknowledging it's possible to sell a car without a title (although that mostly seems to be for cases where the title document has been lost), it does also go on to say (after noting that you should "be especially wary of a junkyard that doesn’t care if you have a title in hand"):

The only exception for the need to produce a valid title is if the state where you live in puts a limit on the need for a title if the car is over a certain age. You’ll still need to produce valid picture ID for the transaction to be legal. This exception is usually made because some abandoned cars might have been left for years in remote locations, and the property owners can find no paperwork for them. In any case, it’s smart to inquire with the local department of motor vehicles if you’re not sure of the regulations regarding sales to junkyards and scrap yards.

So there may be situations where not removing the lien isn't a problem, but overall, as noted elsewhere in comments, the OP's best course is probably to hang on to the paperwork until Covid restrictions ease sufficiently for them to print out the necessary forms and pay the $20 to have the lien removed.