By the evening I'll have done the work
The way I see it this sentence is telling you something about this evening. Maybe you are making plans for the evening or are anticipating how you'll be feeling then. It means pretty much the same as
(By) This evening I'll have already done the work. As such I think 'By ...' can only be followed by verbs which describe states not actions. eg. 'By... I'll be...', 'By... I'll have done...' but not 'By... I'll do...'

If you turn the sentence round then it can be understood in one of two ways. Either as before or as the answer to the (imaginary) question 'When will you have finished the work?'.
If you are talking about something you've already begun or some kind of lenghty process then the perfect will still be the natural choice. However suppose that the task itself is not the problem but the fact, say, that you have so much else to do and might not be able to get round to it. Here, conversely, you would tend to use the simple future. eg. you might say something like 'I'll do it sometime this afternoon' which might easily become 'I'll do it (sometime) before closing-time','I'll do it by closing-time.' You will hear both future and future perfect here.