I know people are going to be searching for this, so until I add some to the dictionary I figured I'd give some examples here.

Tom Lehrer's example requires the Father-Bother merger, as well as enjambment (splitting a word at the end of a line) :

eating an orange
while making love
makes for bizarre enj-
oyment thereof

Some people use door hinge, which requires H-dropping, the Cot-Caught merger, and putting the stress on 'door' when it otherwise might not be. An example from Marian Call:

They drank all the pilsner and ate all the oranges
And she thought as the rotten and rusty door hinges

A perfect one (requiring only enjambment, but no particular features of the speaker's accent as far as I know) would be something worn with a very fancy kilt, 'sporran j-ewel'. Some other perfect ones I found online are Gorringe, Blorange, and Sporange.

I once rhymed 'orange S' with 'foreign ges-ture' in a poem about grapheme-colour synaesthesia.

Do you know of any more?

In the context of programming, "core engine" can rhyme with "orange in". Eg:

I labeled each part of the technical chart; the core engine's orange in color.

Quote from: glenra on 30 October 2016

In the context of programming, "core engine" can rhyme with "orange in".

Note that this one, like 'door hinge', requires the Cot-Caught merger.