You can still carry the same amount of weight - just not as far or as comfortably. (In fact, it may be so uncomfortable that around the block may be "as far.") The stays are what allows you to transfer the weight to your hips. Since your hips and legs are stronger than your shoulders or arms, they can carry the weight longer and more comfortably.

Now, whether or not that's a problem depends on how heavy your load is. With loads under 20 pounds (total, not "base weight" without food and water), you can get by with a frameless pack that doesn't have stays (or from which the stays have been removed.) Over that, you need more support - how much more depends on how much weight you're carrying. Some can carry 20 pounds in a frameless pack. (You'll need to learn how to pack such a pack if you try this. Otherwise, you end up with a pack that rides like a tired three-year-old on your back.)

Before you remove the stays, look at the rest of the suspension. Is there a framesheet that will still remain in place? If so, you're probably OK up to 25 or 30 pounds; you'll still need to pack carefully. Over 30 pounds, you may end up with an uncomfortable carry if you remove the stays.

Another problem in removing the stays is the size of the pack. If a pack is designed for heavy loads, it will have stays to deal with the anticipated heavy loads. But, that usually means the pack bag will be large, too. Removing the stays because you're carrying a lighter (and usually more compact) load may mean that everything ends up in the bottom half of the pack, and therefore not distributed properly. You may be able to make the pack smaller using compression straps, but it will be difficult to size the pack properly.

So, generally, I would opt for buying the right pack for the load I'm carrying - it will probably be lighter than a too-large pack with the stays removed (it won't have all that extra pack fabric), and it will probably be easier to pack and ride better. But, if the budget considerations made it an attractive option, and you can reinstall them if it doesn't work, I'd say go ahead and try.