If you want to save the bulbs so they come back and flower again next year, you should just leave them alone. You should also keep them well watered while they are in flower & have green leaves. Once the leaves fall & go yellow & brown, then you can store them someplace . Keep them outdoors someplace. They need to go through the cold winter.

Another option would be to bin it & pot up some new bulbs in the fall.

You don't need to keep the pots on display once the bulbs have died back - they can go in a shed or garage until about September, when you can repot them with fresh compost. If you want them to flower again next year you should feed them after the flowers have finished, keeping the compost moist but not wet, until the leaves have died back, because that is the time the bulbs build up strength for next year.
If you don't want to keep them in their pots let them dry off after the leaves have died back, and then you can store the bulbs in a cool dry place in paper bags (not plastic or they may go mouldy)

Personally unless you need the pot for something else I'd leave the bulbs in them, you are then more likely to remember them come autumn!

Agree pots would be best but keeping on the right side of Mum matters too!

Hmm, Steragram, not so sure about that, not if she's just plain wrong, though a bit of tact sometimes goes a long way.

Keep them in the pots, out of the way - if you have space outdoors, the hyacinths might as well be put in the ground, and you can plant the tulips out too, now, just plant them a good 4 inches deep. If nowhere to plant outdoors, stand the pots outside or just keep them inside if you have no garden or outdoor area at all.

If they're large pots and the bulbs have died back completely, plant a pelargonium or a pansy or a bizzie lizzie in the top above the bulbs and let that grow and flower till end of summer, that way Mum won't be complaining. Its okay to water the pots if you're growing a plant in the top, just don't let them get waterlogged and don't leave them sitting in an outer tray or pot full of water, empty it after 30 minutes.

Note that forced Tulips rarely bloom the following spring, since the shallowly planted bulbs tend to break up into smaller, non-blooming bulblets, which then take a few years to build back up to blooming size, when they are planted at the proper depth.
Forced Hyacinths are a different story, as long as they were allowed to keep their foliage long enough. They frequently bloom again, though the flower spikes are more slender and graceful than the ones they produced when they were forced. The thick columns are the result of the grower picking out the the nascent spikes before they bloom, over 2-3 years before selling.