Can you skip dialogues in Cave Story?

As far as I can tell, no.

Both Cave Story and Cave Story+ actually attempt to make this more bearable. If you've already sat through a dialog segment and died, the dialog will appear much faster the second time — i.e., box by box instead of scrolling text. These can be skipped through relatively quickly by hammering Z.

Skipping these sequences entirely does not seem to be possible, however.

The v1.1 update (Aug 3, 2017) to Cave Story+ on Switch added an option to skip cutscenes by holding X.

Holding down X during any dialogue scene will cause the game to run in a fast-forward mode. This will speed up all dialogue and animations with an increased framerate.

Currently the option to skip cutscenes is only available on Cave Story+ for Switch. It is unknown if Nicalis plans to bring this feature to other platforms.

Sources

• Reddit: Cave Story Switch Cutscene "Skip" Added in Latest Patch through Uncapped FPS
• Perfectly Nintendo: Cave Story+: Software updates