If the name Oersted actually existed in Danish, which it doesn't, it is anybody's guess how it should be pronounced and what it would mean.
It would definitely be different from Ørsted, which is the name of several villages and, per derivation, a number of humans, some of whom have gained international fame: scientist Hans Christian Ørsted, jazz musician Niels-Henning Ørsted Pedersen, bicycle racer Hans-Henrik Ørsted.
Most likely it would be a three-syllable name: O-er-sted.
The measuring unit
There is one context where Oersted and Ørsted actually do serve as legitimate alternative spellings of the same word: the measuring unit for magnetic field strength.
The rationale most likely is that the ø is interpreted as a German umlaut, ö, which the Germans - unlike Danes - often write oe.