Which game first introduced the concept of weapon recoil?

The first shooter to display weapon recoil, that I have managed to find, is the Dirty Harry (1990) video game.

The game is a side-scroller in which players must guide Dirty Harry throughout San Francisco. He wears a blue suit, although it can be exchanged for a white suit. He wields his signature Smith & Wesson Model 29, and players have the ability to draw the weapon without actually firing it. The Smith & Wesson Model 29 also exhibits a recoil effect when fired.

To fulfill the FPS requirement, Time Crisis (1995) might be your best bet as it was an arcade game so the point of view is your own eyes. I know from personal experience that the arcade machine forced recoil.

Most sources tend to be relating this back to the police Fire Arms Training Simulator (FATS) training systems, like this article does, pg 15::

Now we use large screen TVs and soldiers stand with plastic M-16s that fire laser beams that when you hit the target on the screen, the target drops. The law enforcement community extensively uses a device known as the FATS trainer: Fire Arms Training Simulator. You hold the gun in your hand, you pull the trigger,the slide slams back, you feel the recoil, you hit the target, the target drops, you miss the target, the target shoots you. It is a very effective law enforcement training device.

But if you go to the local video arcade, you’ll find an almost identical device. A game such as Time Crisis for example, in which you’ll find that the pistol, the slide slams back, it recoils in your hand, if you hit the target the target drops, if you miss the target the target shoots you. The only difference is in the FATS trainer, if you shoot the wrong target, you’ll be reprimanded, ultimately even fired.But when the kids are playing the game there is no adult supervision, there is no standard, there is no control.

Or mass murder performed by children, as this article does:

• Silent Scope has a mounted sniper gun with a scope. You sneak up on people and shoot ordinary people for no reason. When you kill, blood splatters everywhere. You get extra points if you shoot your victims in the head.
• Time Crisis and Time Crisis 2--This has a realistic recoil action gun. Guns make sounds like real gun sounds. It is 3D.
• Mortal Kombat series, Mortal Kombat Ultimate--This has joysticks. You use your fists and legs and feet. Bodies explode blood when you hit them. Mortal Kombat Ultimate says on the screen--``There is no Knowledge that is not Power.'' Does that mean that if you know how to kill someone, then you will have power?

More of an RPG with FPS elements than a pure FPS, but the original System Shock from 1994 had projectile weapon recoil.

This full-sized combat rifle is based on the 2064 Interlocutor KR-5. It's not quite as powerful as it sounds, and it produces a big recoil and thus has a longer recovery time than the other weapons. Still, it's handy to have around, and can dispatch most enemies with a couple of shots. Penetrator ammo is very scarce, but highly effective.

According to Phoenix: The Rise & Fall of Video Games, 3rd edition :

In 1989 :

Konix developed other controllers for the system. One was a light gun that had attachments to turn it into a light rifle. It even had a recoil when it fired to make it realistic.

According to wikipedia, the system with recoil was first demonstrated in February 1989 at Earls Court Exhibition Center in London.

(but this was physical recoil, not animated recoil)

In the March 1984 issue of Crash: Micro Games Action, page 25, the game "The Pyramid" is reviewed and the review says:

there is even a recoil effect on the laser

(also if you consider military simulators, already by World War II there were recoil simulators for automatic weapon training)