Yes, sure, if your gig is out of search for 10 days, just as if you turn on Out of Office mode, that would lower your gig’s search position. Fiverr has clearly stated to multiple people who posted that info on the forum, that their focus is on presenting gigs that are active, thus, logically, gigs that aren’t active, will fall in the ranking.
Your gig will go up again once you’ll deliver again, but how long that will take depends not just on your gig but also on the gigs of your competitors, of course. The ranking is based on a number of factors.
To simplify it, let’s say your gig was out of search for 10 days and on day 11, you deliver, competitor 1 had Out of Office mode on for 5 days and delivers the first order after that on day 12, competitor 2 had his gig in search all the time but got overbooked and delivered 2 orders late, competitor 3 had all gigs in search all the time and all stats are good but they got an account warning for a ToS violation, competitor 5 had her gig in search all the time but had 2 cancellations.
Hard to say whose gig would rank in place 1, 2, 3, 4 and 5, right?
What you could do to mitigate the “damage” of having to pause/limit your gig for too long, is to have a gig for something related, same category/niche as your main gig but for something that takes you so little time that you could do a few of those gigs during those (10) days, and put in its description that you also do (what your main gig offers) and to please contact you for that; then people who are interested, can still contact you about that gig and if they are interested enough, maybe they are fine to wait 10+ days. You could send them a custom offer for, e.g. 15 days, you can still send custom offers even when the limit of the gig is reached.
I hope that helps a bit. Generally, you need to keep in mind that Fiverr’s aim isn’t as much to present every sellers’ gigs constantly but to present gigs to potential buyers that are available. So, yes, if your gig isn’t available, there is not much motivation for Fiverr to present it in a high position. Once it is available again, it should climb again, the more, the sooner/more deliveries you make.
ETA: In scenarios like yours, it’s very beneficial to have regular customers, so you’ll have better chances of making deliveries soon after your gig is in search again.
If you don’t have regular customers, it might be good to consider if you can offer something as a gig that would attract people who’d need your service multiple times/regularly - as then you’d even get orders when your gig ranks low(er), and the delivery of those orders should help you regain your gig position faster.