The Earth Will Explode If Gohan
Turns Super Saiyan
This rumor possibly goes back as far as we have been involved in this fandom; its origins are completely unknown. In a pre-Dragon Ball Super era, fans always looked for a reason as to why Gohan did not become a Super Saiyan after receiving his so-called “Ultimate” power-up from the Old Kaiōshin. However, there is nothing in the series that implies that the Earth would be in any type of danger if he did go Super Saiyan again.
A much more logical explanation would be that after he gets powered up, he does not need to become a Super Saiyan anymore. With all of the latent strength within him awakened, he would simply be using up valuable ki by doing so. Nevertheless, the argument of this rumor, that the earth would explode if Gohan went Super Saiyan after receiving his power-up, is and always was completely baseless.
Nothing about this rumor really changes with the inclusion of Dragon Ball Super material, where the series actively explores Gohan’s continuous cycle of training, not training, re-focusing his efforts, re-discovering his own path, etc.