I just ran into the same problem.
I have just encountered the same problem

However, now I’m getting a new error, about nested processes, and mlflow wanting me to set nested=True in the start_run call that is embedded in the MLflowCallback class

And found out the cause of this error.
The cause of this problem was that Trainer’s __init__ automatically added a callback based on the installed package.
As a result, the callback given as an argument to Trainer was duplicated and mlflow.start_run() was executed twice.
The solution is to not specify the mlflow callback as an argument.
trainer = Trainer(
    args=training_args,
    tokenizer=tokenizer,
    train_dataset=train_dataset,
    eval_dataset=eval_dataset,
    model=model,
    compute_metrics=compute_metrics,
)

As a side note, I found the cause in lines 391-395 of trainer.py.
get_reporting_integration_callbacks(self.args.report_to) returns all integration callbacks installed and added it into callback handler.
default_callbacks = DEFAULT_CALLBACKS + get_reporting_integration_callbacks(self.args.report_to)
callbacks = default_callbacks if callbacks is None else default_callbacks + callbacks
self.callback_handler = CallbackHandler(
    callbacks, self.model, self.tokenizer, self.optimizer, self.lr_scheduler
)

If you’ve already solved it, I’ll leave the solution for those who are facing this problem anew.