System Information
Operating system: Linux (Ubuntu 20.04)
Graphics card: GeForce GTX 650 Ti

Blender Version
Broken: 2.90 beta (26 august)
Worked: not 2.83, maybe before?

Short description of error
When animating a shape key's Max Range, playback in the 3D viewport behaves as expected but not the rendered animation. Whichever value is at the current frame in Max Range at the time of starting the render gets used for the entire duration, regardless of the inserted keyframes.

Exact steps for others to reproduce the error

1. Open Blender and add two shape keys to the cube (Basis and Key 1)
2. Edit the cube mesh for Key 1 to be different from Basis
3. Add a few keyframes for the Value of Key 1, like say, frame 1=0, frame 25=1, frame 50=0, and frame 75=1
4. Keyframe Key 1's Range Max to go from 1 to 0 (frame25=1 and frame50=0)

• Press play in the viewport:
  • At frame 25 we have the Key 1 shape and at frame 75 we have the Basis shape
• Place the playhead at frame 1 then render the animation:
  • Key 1 shape at both frame 25 and 75
• Place the playhead at frame 50 (or later) then render the animation
  • Basis shape at both frame 25 and 75

Blend test file