If the creature is unwilling then nothing happens
Dnd 5e operates on the principle that things only do what they say they do. Jeremy Crawford, lead rules designer for 5e, and known for adjudicating rulings on Twitter, has stated:
Beware of claims that a rule does something mentioned nowhere in that rule or elsewhere in the core books. There aren't secret rules. (source)
See this question, for further reference.
Following this principle, if the designers had meant a School of Conjuration Wizard to be able to swap places with an unwilling creature, subject to a saving throw, they would have stated so in the ability description. So, RAW swapping places with an unwilling creature, using this ability is impossible.
Further support for this conclusion has been suggested by mxyzplk, in the comments:
...this is why the word 'Benign' is in the name. In 3.5 there is in addition a 'Baleful' Transposition that does operate on an unwilling subject.
And, if you try, you'll probably lose your action.
If a PC wanted to try and do this, then it would be up to the benevolence of their DM as to whether they either simply "realise on reflection that such a course of action is impossible", or they lose their action that turn by trying to do it unsuccessfully.
Xanathar's Guide to Everything is an optional rules source, but it provides details on what would happen when a player tries to Cast a Spell on an invalid target.
Invalid Spell Targets (XGtE p. 86)
... If you cast a spell on someone or something that can't
be affected by the spell, nothing happens to that target,
but if you used a spell slot to cast the spell, the slot is
still expended. ...
Benign Transposition is a class ability rather than a Spell, but it seems logical that a failed use of it would follow a similar pattern - your action and one usage of the ability would be consumed, despite the failiure.
Could you 'homebrew' transposing an unwilling creature as possible?
In your own game your DM is welcome to rule entirely as they like, and in that case maybe a Charisma save might be argued to be appropriate for the unwilling creature (similar to the spell Banish)? However I would personally advise against this ruling. I think it would add a great deal of power to this ability and probably unbalance it.