Sure, all you need to do is make sure the problem_type of the model’s configuration is set to multi_label_classification, e.g.:
from transformers import BertForSequenceClassification

model = BertForSequenceClassification.from_pretrained("bert-base-uncased", num_labels=10, problem_type="multi_label_classification")

This will make sure the appropriate loss function is used (namely, binary cross entropy). Note that the current version of Transformers does not support this problem_type for any model, but the next version of Transformers will (as per PR #14180 1).
I suggest taking a look at the example notebook 10 to do multi-label classification using the Trainer.
Update: I made a notebook myself to illustrate how to fine-tune any encoder-only Transformer model for multi-label text classification: Transformers-Tutorials/Fine_tuning_BERT_(and_friends)_for_multi_label_text_classification.ipynb at master · NielsRogge/Transformers-Tutorials · GitHub 3