No, it's a lie. Consumers paying more because of data centers raising demand could be true, but that's not equivalent to them paying for the data centers' usage. The data centers also have to pay an increased rate when prices go up.

Data centers get commercial or maybe even industrial rates depending on their grid hookup and utilities love predictable loads. Those are lower than residential rates. If you're dishonest and don't understand the cost of operating a grid, you could say that's users paying for data centers. But then you'd need to apply it to every commercial/industrial user.

If the regular users were paying for data centers usage, why are so many of them going off-grid with turbines or at least partially on-prem generation?

The solution is more and cheaper energy.