Why did the Old English word "līċ" get displaced by "corpse"?

What Happened?

Lich in Old and Middle English referred to bodies in general. In the OED's two definitions for lich, that body could be alive ("lich, n.," def. 1) or dead ("lich, n.," def. 2). That's important to understand because corpse could also refer to a living or dead body ("corpse, n.," def. 1 and 2).

Between Middle English and Early Modern English, two changes occurred:

1. Lich dropped out of usage, except in specialized combinations like the lich-gate or the lich-house.
2. Corpse increasingly pertained to a dead body, rather than to either a living or dead body.

When Did Lich Decline?

Corpse (in earlier forms like cors) was in Middle English by the 13th century, according to the Middle English Dictionary, and stays in the language from then on:

c1275 Ken.Serm.(LdMisc 471)216/55 : Mirre..defendet þet Cors þet is mide i smered þet no werm nel comme i hende.

Lich, in contrast, is swiftly disappearing by the 16th century, appearing primarily in printed editions of older manuscripts, like this (printed 1600 from a text dated to around 1350):

c1600(c1350) Alex.Maced.(Grv 60)195 : With likand legges lovely too seene..Lili-white was hur liche.

The usage may have stuck around in spoken language for much longer, but usages in print after this point are intentionally archaic, like this 19th century ballad:

1806 Sir Oluf in Jamieson Ballads I. 222 Three likes were ta'en frae the castle away.

Why does this replacement occur?

One guess is that speakers favored the more clinical, educated sound of the French-derived cors/corpse over the common use of lich. Such lexical substitution was common in the Middle English period due to the influence of England's Norman aristocracy (Wikipedia).

Another is that lich had an uphill battle because it looked the same as a very common word, like. The lists of forms for lich and like look rather alike; both could appear as lic, lich(e), lych(e), lyke, and like. They may have sounded the same too. So it's also possible that lich underwent a decline in usage compared to a similar-meaning word (corpse) in order to avoid confusion. At around the same time, corpse specialized to fill its current semantic niche.