Q:

What do you call a bug collection?

Entomology is the study of insects, which can informally include minibeasts that aren't scientifically classified as insects. Professional and amateur entomologists sometimes maintain bug collections.
Is there a single word for such a collection - a curated assortment of various species of bugs? Kind of how:

a beekeeper keeps bees in an apiary
animals of various sorts are kept in a menagerie or zoo
taxidermalogical mounts might be presented in a diorama
a philatelist collects stamps in an album

So, an entomologist keeps bugs in a(n) __________ .
Notes:

Implying that the insects are living or dead isn't required. (Either implication is fine.)
Implying professional or amateur interest isn't required. (Either implication is fine.)
A phrase is fine, as long as it doesn't include the synonyms of "insect/bug/minibeast" or "collection". Because then "bug collection" would do :)

A:

An insectarium or insectary is, per Oxford Living Dictionaries,

A place where insects are kept, exhibited, and studied.

In practice, web searches suggest that insectarium is preferred for institutions which function as zoos for live insects, whereas insectary refers to the raising of insects, whether for institutional research or to attract beneficial insects to one's garden.
For example, some institutions for public education include The Insectarium, a component of the Space for Life (Espace pour la vie) in Montreal; the Philadelphia Insectarium and Butterfly Pavilion; the Audubon Butterfly Garden and Insectarium in New Orleans; and the forthcoming Insectarium of Victoria, about an hour outside Melbourne. Insect zoo is another term for these public-facing collections, like the Smithsonian Institution's O. Orkin Insect Zoo or the Insect Zoo at the Portland Zoo.
On the other hand, facilities where insects are raised primarily for research seem to favor insectary, from the U.S. Department of Agriculture to the Colorado Department of Agriculture, to the entomology or agriculture programs of many U.S. research universities.
Collections primarily of dead insects kept for research or display may simply be referred to as insect collections, like the Iowa State Insect Collection or the Cornell University Insect Collection.

The -arium or -orium suffix, borrowed from Latin, is often used to produce words about a place associated with something, like aquarium or auditorium which may be familiar to you. A xylarium is a wood collection, a formicarium is a pretentious ant farm.

A:

An earlier answer, insectarium for living insects, satisfies your parallels with living bees (an apiary) and animals (a zoo or menagerie). That is, 

insectarium, n.
  A place for keeping and breeding insects; an entomological vivarium.  
OED

The insectarium, however, is a curated collection of living bugs. 
For displays of collections of dead bugs (entomotaxy), two terms satisfy your parallels with taxidermy (diorama) and philately (album). Those terms are combinative (just as are taxidermological diorama and stamp album), but in the appropriate context the terms may be used to good purpose without prepending the combining term insect: 

insect-box, n.
  ....
insect-cabinet, n. 
OED

As suggested by the usual meanings of box and cabinet, the former is usually associated with a smaller scale, more temporary display and arrangement of bugs (or a subdivision of a cabinet), and the latter with a larger and more permanent display. 
Here is a general description of an insect-cabinet, from Directions for collecting and preserving insects (A.S. Packard, 1873): 

Insect-cabinet. For permanent exhibition, a cabinet of shallow drawers, protected by doors, is most useful. A drawer may be eighteen by twenty inches square, and two inches deep in the the clear, and provided with a tight glass cover. For constant use, baxes made of thin, well-seasoned wood, with tight fitting covers, are indispensable. For Coleoptera, Dr. Leconte recommends that they be twelve by nine inches (inside measurement). For the larger Lepidoptera a little larger box is preferable. Others prefer boxes made in the form of books, which may be put away like books on the shelves of the cabinet, though the cover of the box is apt to be in the way.

The style and details of insect-box and insect-cabinet construction and appointment vary widely.

