Pre-trained transformer language models have shown remarkable performance on a variety of NLP tasks. However, recent research has suggested that phrase-level representations in these models reflect heavy influences of lexical content, but lack evidence of sophisticated, compositional phrase information. Here we investigate the impact of fine-tuning on the capacity of contextualized embeddings to capture phrase meaning information beyond lexical content. Specifically, we fine-tune models on an adversarial paraphrase classification task with high lexical overlap, and on a sentiment classification task. After fine-tuning, we analyze phrasal representations in controlled settings following prior work. We find that fine-tuning largely fails to benefit compositionality in these representations, though training on sentiment yields a small, localized benefit for certain models. In follow-up analyses, we identify confounding cues in the paraphrase dataset that may explain the lack of composition benefits from that task, and we discuss potential factors underlying the localized benefits from sentiment training.