Do noble deeds, not dream them all the day.
Do noble deeds, not dream them all day.
Can these two sentences be used interchangeably? If not, which one is correct?
I believe both sentences are correct and mean the same thing. I assume that all day is a more common expression than all the day.
Yes, Google Books confirms it.
In addition, there is a very good answer on ELU given by @Sven Yargs. I will cite the paragraph which summarises the difference between these two in modern English.
In contemporary idiomatic English, however, "all the day" is something of an oddball phrasing—certainly far less common in everyday speech than "all day"—for reasons that may have more to do with random drift than with any systematic change in informal English usage. In any event, you can use "all the day" without fear of being misunderstood by others, but the usage is likely to mark you as a non-native speaker or, perhaps, a folksinger.
The original quote by Charles Kingsley also has long at the end, which slightly emphasises all day.
Do noble things, not dream them, all day long.