Some attorneys say so (e.g. https://www.avvo.com/legal-answers/is-the-discovery-process-halted-when-a-motion-for--3637798.html), but when I looked into the statute (CCP 2030.090, 2031.060, 2033.080), I never found anything that "stays" the discovery pending the hearing of motion for protective order, or explicitly states that the moving party does not ned to comply with the discovery until the motion is ruled.
This is especially important for "request of admissions" because the lack of response would enable the propounding party to move the Court to deem every request as if they were admitted.
Lack of response to request for production of documents / interrogatories also almost guarantee a compel / sanction motion to be granted.
Which means, if the party seeking protective order just want to be safe, they would need to respond to the (supposedly unreasonable) discovery anyway, even if the response is just a whole bunch of objections. Moreover, some objections have to be very specific (e.g. exactly why something is over-burdensome, what effort would it take, etc.), which means it would be a lot of work to formulate a bullet-proof response. Wouldn't that defeat the whole purpose of motion for protective order?