If the beacon block is beyond render range vertically, the beam does not render until the player gets within render range vertically. The player can they go above render range and the beam remains. On a server with server render distance 12 beacon beams will not render if a player does not get within 198 blocks vertical, even if the player render distance is set higher. If the player sets the render distance lower the beam will not render until the player gets as close as their client render distance. Screenshots attached with server render distance 12 and client distance 32. Tested by logging in at different y levels and seeing if the beam would appear. Once the beam was rendering I could go well above and the beam would remain.

I would expect the beacon beam to always render if the beam passes through render distance, vertical distance from the beacon block should not matter. The game knows to apply the beacon effect, but the renderer doesn't know to render the beam.