My version (Scribner) has a slightly different quote:
These people! You have to keep after them all the time (36).
However, there is irony in that statement, too.
To see the irony in either statement, it is important to understand what irony is first. Irony is a juxtaposition of incongruous elements, in other words, putting things together that do not fit.
Next, we need to examine the context. Myrtle is clearly of a very low socio-economic group, much lower than that of Nick, Tom, and Daisy, who are, I would say, upper-class. Myrtle in this chapter is drinking heavily.
Now, given your quote, the irony is that someone who is drinking is talking about how someone else needs to be drunk all the time, in order to perform services properly.
The other irony, which is present in both quotes, is that Myrtle is acting as though she is of a much higher class than the elevator man, putting on airs, as they say, and in fact, it is likely that she is of the same class, or perhaps even lower.