Yes, intrusive interstitials for mobile searches can have a negative impact on your search engine rankings. On January 10, 2017, Google announced that is taking action against websites that displayed intrusive interstitials:
“Starting today, pages where content is not easily accessible to a user on the transition from the mobile search results may not rank as high. As we said, this new signal is just one of hundreds of signals that are used in ranking and the intent of the search query is still a very strong signal, so a page may still rank highly if it has great, relevant content” – source
The same applies to app install interstitials.
But, can newletter subscription pop-ups negatively impact rankings, as well? In Jan 2021 in a hangout we heard the following question:
… Because there is actually one American paper that, like, literally every day, a pop-up comes to sign on to its newsletter. You can see like a little of the text below and under, but you have to click it every time, and I’ve actually wondered about, […] are they really getting that many people on a daily basis to sign up for this newsletter compared to what this might be doing negatively to them? Is that something that would be a negative? Because it is, I think, extremely annoying.
https://youtu.be/MGAedxutADU?t=1294
To which, John Mueller replied:
That sounds like something that we would pick up on. One of the things there is that we focus on the mobile version of the site, and we only use that in the mobile search results. So, that’s something where if they’re not showing it on mobile, then maybe we wouldn’t be picking up on that. But if it’s on mobile as well, then that would definitely fall into the category of intrusive interstitials, and we’d say this is kind of like something where we would slightly demote the website in Search. Sometimes the tricky part there is with the slight demotions. It’s not the case that we’ll remove the site from Search, or we’ll kind of like move it to page 100, or something like that. But if it’s really relevant content, then maybe we’ll still show it on the first page of the search results, just not like as highly as it could be.
https://youtu.be/MGAedxutADU?t=1294
So, yes, if those newsletter sign-up modals are not implemented properly, they can have a negative effect on rankings.