First of, you never cold storage spring bulbs over summer. They do need a cold period in winter to make them bloom and in very warm areas one will need to artificially chill them, but that doesn't apply to your area as far as I know.
If they need to be lifted they should be dried airy and preferably above room temperature and then also be kept relatively warm over summer, the optimal temperature depending on the species.
Other than the occasional species requiring high summer temperatures to initiate bloom or to bloom more freely in cold summer climate such as my own here in Sweden (I suspect that this isn't really a problem in most areas of the US) lifting bulbs is something you do to keep them healthy, dry and happy over summer. This is especially true if you plant them in an area that has moist soil over summer.
In the case of tulips you would also lift them simply because some of them divide so quickly that they will break down into a crowded clump that will not bloom. I do lift a lot of my tulip bulbs as they benefit greatly from it, but I agree 100% with Lilli that it isn't necessary for all cultivars, but it also depends on your soil and how dry and well drained it is.
Hyacinths on the other hand divide slowly so do not need to be lifted very often. In the past I have never lifted them for summer, although I have recently begun to do so, in part because I'm curious and enjoy to see how they develop over the years, but also because some cultivars haven't been as perennial as I would prefer. I would suggest to try some of the more common colors. Especially the dark blue are very perennial here, but some of the more 'exotic' colors, such as yellow and apricot a little bit less so and has only survived well in a very dry spot. This doesn't mean they wouldn't do well for you, but it is something to keep in mind.
The common Muscari armeniacum is one of the more tolerant bulbs and I have personally tried growing Muscari aucheri 'Blue magic', 'White magic' and 'Ocean magic' as a ground cover below daylilies with no ill effects on the bulbs other than it is a mess when they or the daylilies need dividing. These increase rapidly, but unlike most tulips they bloom even in a quite crowded clump.
BTW spring blooming bulbs do not generally appreciate to be mulched over summer, so if you use mulch I would suggest not to apply it to where you have bulbs.