I like it as tabs as it gives a fast way to look at closed issues assigned to me at Dashboard / Issues. Also I believe for small projects open/closed filtering is primary one to use.
I am strongly against it. And I agree with @dzaporozhets's arguments.
Another major reason: Now that #15356 (closed) got fixed (coming in %8.13), the number badges in the tabs will actually show the number of open/closed issues/MRs after filtering. That is a big deal as it gives you immediate (and extremely valuable) feedback to your filters/search.
It helps finding issues quickly (example: i have a pretty good idea what I'm looking for but the issue got closed a few minutes ago. Applying filters show 0 open issues but >0 closed ones.)
It helps getting an idea how many issues concerning one topic/component exist in total (currently impossible, as the counters don't change)
It helps avoiding the creation of duplicate issues (example: searching for a specific error message gives 0 open issues but 1 closed one)
In other words: Seeing the count for both open and closed issues/MRs at a glance is helpful. A dropdown doesn't allow this.
Yes, I agree with both @dzaporozhets and @pbr. I think the state of the issue makes sense as tabs, not a dropdown. Especially because of the point about viewing the number of open/closed issues/MRs after filtering.