Can a gas-fired burner and an oil-fired burner be connected to the
same residential chimney flue?

My not-professional opinion is yes, however it may be prohibited by
some local ordinaces. It likely is not wise to use a flue for a fire
place for other purposes. In one house a downstairs fire place dumped
smoke into the house though the up stairs fire place, but the problem
likely would have been rare if both fire places had been on the same
level. Most modern houses are tightly sealed, so both oil and gas
burners are a significant hazard even with well designed independent
flues, especially when a clothes dryer is venting inside air to the
outside.   Neil