Hello ScottyUK and edison123
I agree with you scottyUK, with three exceptions.
A single phase transforme with the secondaries in parallel should have no circulating current. If however, the secondary windings are not perfectly balanced there will be a small circulating current. This will add a small loss that will be removed if the winding is removed.
If the secondary was delta connected and was carrying any harmonic currents, there will be a slight reduction in losses if the secondary is removed.
If the primary is 4-wire wye connected and the secondary is delta connected there are excessive currents when the primary voltage is unbalanced. The circulating currents and the resulting losses can exceed the full load losses. (That's probably why the RUS standards warn that this connection can result in transformer burnout.)
I understand that some three-phase transformers use a core configuration that results in harmonic fluxes that need a delta winding to provide a return path. Hence tertiary windings.
I'm open to comments and correction on this point.
For most transformers you are correct scottyUK. I believe that for all single-phase transformers a series connected secondary has no effect on the no load losses.
For three phase Wye Delta transformers, depending on the core arrangement, harmonic currents and the primary voltage balance, the removal of the delta winding, (Whether secondary or tertiary), may result in the same losses, less losses, a lot less losses, or more losses.
The definitive word, to quote one of our other contributors, is "It Depends".
respectfully