Will changing host hurt my search rankings?

No, not by itself. Search engines rank URLs based on their content, links and technical health, and a straightforward migration to new hosting changes none of those things — the pages are the same pages, at the same addresses, just served by a different server. What actually damages rankings during a migration is downtime, broken pages, or URLs that stop resolving, and all three are avoidable with the same order of operations that prevents downtime in general.

What genuinely does put rankings at risk

• Extended downtime. A site that returns errors for an extended period can be temporarily removed from search results. A brief overlap during DNS propagation is not this — search engines tolerate short blips without penalty.
• Broken pages after the move. If a folder was missed in the copy, or the database import was incomplete, pages that used to work start returning errors. Search engines eventually recrawl and notice.
• URLs that change without redirects. This is not caused by moving host — it is caused by also restructuring the site's URLs at the same time. If old page addresses stop working and nothing redirects visitors or search engines to the new ones, that traffic and ranking history is genuinely lost. See keeping your URLs working after a move.
• A slower site on the new host. Page speed is a minor ranking factor. If the new server is genuinely slower — wrong PHP version, missing caching, an under-resourced plan — that can have a small effect over time, though it is a hosting quality issue rather than a migration one.

What a clean migration looks like from a search engine's point of view

If you follow the standard order — copy the site, test it fully on the new server, switch DNS, keep the old host running for a week — a search engine crawler experiences almost nothing unusual. It requests the same URLs it always has, gets the same content back, and at some point notices the responses are coming from a different IP address, which is not something it treats as significant on its own.

Do you need to tell Google you moved host?

No, not for a like-for-like host change with unchanged URLs — there is nothing to tell it, because nothing about the site's addresses or content changed. This is different from a domain change, which does need to be declared through Search Console. See do I need to tell Google I changed hosts for the distinction.

What to actually do about it

Follow the standard migration order, test thoroughly before DNS switches using a proper pre-launch checklist, and keep the old hosting running for about a week afterwards. Do that, and the migration is invisible to search engines in exactly the way you want it to be.