To answer my initial question, the answer was that I needed to put the MLFlowCallback class inside a list when specifying the callbacks parameter:
trainer = Trainer(
    ...
    callbacks=[MLflowCallback],
)   

I also noticed that the MLflowCallback class calls mlflow.start_run() itself, so no need to call it myself in my script. So new code looks like this:
import torch
from transformers import RobertaTokenizer, RobertaForSequenceClassification, RobertaConfig, Trainer, TrainingArguments
import mlflow
from transformers import TrainerCallback
from transformers.integrations import MLflowCallback

remote_server_uri = [PRIVATE SERVER URL]
mlflow.set_tracking_uri(remote_server_uri)

# After loading and tokenizing data, here we run the training experiment
experiment_name = "ht_vp_roberta_randomSearch"
mlflow.set_experiment(experiment_name)             
training_args = TrainingArguments(
    output_dir=experiment_name,
    evaluation_strategy='epoch',
    eval_steps=500,
    gradient_accumulation_steps=1000,
    eval_accumulation_steps=1,
)
model = RobertaForSequenceClassification.from_pretrained("roberta-base")
trainer = Trainer(
    args=training_args,
    tokenizer=tokenizer,
    train_dataset=train_dataset,
    eval_dataset=eval_dataset,
    model=model,
    compute_metrics=compute_metrics,
    callbacks=[MLflowCallback],
)
trainer.train()
trainer.evaluate()

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
Traceback (most recent call last):
  File "hypertune_mlflow_test_simple.py", line 89, in <module>
    trainer.train()
  File "/home/jovyan/conda/dsEnv/lib/python3.8/site-packages/transformers/trainer.py", line 1224, in train
    self.control = self.callback_handler.on_train_begin(args, self.state, self.control)
  File "/home/jovyan/conda/dsEnv/lib/python3.8/site-packages/transformers/trainer_callback.py", line 340, in on_train_begin
    return self.call_event("on_train_begin", args, state, control)
  File "/home/jovyan/conda/dsEnv/lib/python3.8/site-packages/transformers/trainer_callback.py", line 378, in call_event
    result = getattr(callback, event)(
  File "/home/jovyan/conda/dsEnv/lib/python3.8/site-packages/transformers/integrations.py", line 665, in on_train_begin
    self.setup(args, state, model)
  File "/home/jovyan/conda/dsEnv/lib/python3.8/site-packages/transformers/integrations.py", line 641, in setup
    self._ml_flow.start_run()
  File "/home/jovyan/conda/dsEnv/lib/python3.8/site-packages/mlflow/tracking/fluent.py", line 109, in start_run
    raise Exception(("Run with UUID {} is already active. To start a nested " +
Exception: Run with UUID 8a5c8b90de7c412fb6c857b54416f346 is already active. To start a nested run, call start_run with nested=True