When you remove an internal link a page has relied on, the page loses the share of authority and the discovery path that link was providing, and how much it loses depends almost entirely on how irreplaceable that link was. The effect is not instant; it shows up after search engines re-crawl and reprocess the affected pages, which happens over a span of crawl cycles rather than overnight (this timing is observed behavior worth confirming for your own site, not a fixed rule).
The mechanism turns on uniqueness. If the removed link was one of several internal paths to the page, the loss is mild, because authority still flows through the other links and the page stays easy to discover. If it was the page’s main or only meaningful internal path, the loss is sharp: the page may receive less internal authority, get crawled less often because there are fewer routes to it, and in the extreme become near-orphaned. The worth of any internal link is roughly its uniqueness as a path, so a sole route matters far more than one of many.
The timing follows the crawl. Search engines do not notice the removal until they re-crawl the page that used to carry the link and reprocess the link graph. On a frequently-crawled site that is faster; on a rarely-crawled one it is slower. Either way the ranking effect lags the edit by crawl cycles, which is why a link cleanup can look harmless for a while and then show its cost once the re-crawl catches up.
Before any link cleanup, identify which internal links are load-bearing: pull the list of internal links pointing to your important pages and note any page whose discovery or authority leans heavily on a single link. Leave those in place. Cut the redundant links freely, but treat a sole path to a page you care about as structural, because removing it trades a tidy edit for a ranking cost that arrives a crawl cycle or two later.