The threat of eviction adds a dramatic extra bit to the scene and to Ben's overall burden, but I don't see it as the impetus for Ben to pack, nor is the eviction official.
Even if it were an explicit eviction I think realistically Ben would have a few days to negotiate and delay moving out. That he might immediately up and leave is consistent with his character, of course, as a leisurely pushover who reacts to seeming bottleneck situations without consideration for what comes next. However, this same characterization means the scene works on the strength of Ben and Elaine alone. In this case, the landlord is just there for dramatic reinforcement.
Elaine has just delivered a double punch (paraphrasing: "You raped my mother and I want you out of my school!"). Ben's packing can be taken as a tail-tucked pout, a pathetic gesture for her, his audience. He might have gone all the way through with it in his underthought reactionary way, except she turns before leaving and tells him to stay and "figure out a plan first."
The major plot point here is in Elaine's dialogue. She's the current that pushes Ben out to sea, so to speak. The landlord is just a wave in the same direction.
Later that night, Elaine returns to make nice, and Ben doesn't seem to have done any more packing. He does, however, seem to have given thought to leaving -- with Elaine. Again in his underthought reactionary way he proposes marriage. The next day he dogs her for a yes and comes home with bundles of gifts to find Mr. Robinson waiting to deliver his own double punch (paraphrasing: "You made a fool of me and you'll never get near my daughter!"). The landlord repeats his eviction threat.
This time the major plot point is in Mr. Robinson's dialogue; he's now the current, with the landlord a wave again. Since the climax of the film kicks off proper here and Ben goes for broke getting to Elaine, we don't ever return to the house to find out how serious the eviction threat ever gets.
The landlord is serious about the eviction but there is no practical way to stop Ben from coming back. Ben opens the door without a key, both arms full. Mr. Robinson is in the room already and he certainly doesn't have a key. He's also been there "for quite a while" (1:27:38 on the DVD) presumably without the landlord's awareness or approval. The landlord seems surprised by Mr. Robinson's presence. It's possible Ben came home minutes before the landlord visited for the first time that day and they cross paths right at the moment the landlord is checking whether Ben had left for good.