Can I remove the circuit breaker handle ties and treat them as individual circuits?
But they're not individual 120V circuits!

They are combined into pairs, as something called a multi-wire branch circuit.  These share a neutral, so they're not separate at all.  

The handle-ties exist for a bunch of reasons.  

They must be on different poles/phases

Top of the hitlist is to force them into different poles or phases, i.e. To keep someone from obliviously moving them onto a double-stuff breaker so they're on the same pole.  That would set the neutral wire on fire, because the neutral wire would be handling all the current for both hots.   

Whereas, if they are on different poles/phases, neutral only handles differential current: there's some math, but on a split-phase or 2-of-3-phase system, neutral will never have more current than any one hot.  This is the slick trick of the multi-wire branch circuit.  Take a look at a big box store's lighting sometime; they make full use of 3-phase MWBCs at 277V, and it lets them put over 20,000 watts of lighting on twelve #12 wires in a 3/4" conduit.  

Anyway, as a practical matter, in modern service panels, the presence of a mandatory handle-tie forces you to position the MWBC on opposite phases/poles. 

That is its most important function  where someone has an itch to rearrange the panel.  We'll come back to that. 

Common maintenance shutoff

Typically, an electrician plugs a radio into a circuit, and shuts off breakers until the radio falls silent. This ought to work on a MWBC.  Otherwise, the electrician could shut off half the MWBC, and promptly get nailed by the other half.  "But how can that be? He'd only be messing with neutral, and that's harmless!"   If that were true, we wouldn't insulate it... Actually neutral is hot, it's just near ground if everything is working right and properly connected. 

The handle ties ensure that when he silences  the radio, he has shut off the entire circuit.  

This is the part of the handle-tie that is actually Code mandated.  Forcing opposite poles is just a nice side effect.

But my half is tripping because of overuse of the other half!

My home is fantastically underserved.  It is on half an MWBC; the other half is shared by 2 apartments.  (Really.) Even so, having to reset a breaker from overload is a very rare occurrence, (maybe once every 5 years in my lifetime).  It should be rare.   If it's not rare, someone is doing something wrong and not taking a hint. 

The cure for this is to learn about wattage of devices. Most people have no idea a hair dryer takes more power than a 54" TV.  They're plainly labeled: learn what that means, learn what is on which circuits, and don't overload the circuits.   My partner figured it out with only 3 explainings, and knows exactly how to sequence microwave, toaster, hair dryer and various heaters on high vs low. 

But I want to overstuff my panel!

A constant refrain here is "buy a huge panel".  Because it only costs a few dollars more, and it saves you from having panel-cram problems later.  I gather your builder didn't get the memo.  

Our quickest cure for that is "add a subpanel".  I like to see a total number of full breaker spaces  for a house to be in the forties.  So if you have a 24 space, get another one. 16, get a 30. Etc.   There are other strategies for underserved or obsolete/dangerous panels, that involve preparing the subpanel to become the new main panel when resources permit a changeover.  This isn't spendthrift, this is a very economical way of being smart. 

However, if you are double-stuffing this panel, here is the takeaway:  they have quadplex breakers with a 2-pole breaker in the center and two individual breakers or another 2-pole on the outside. MWBCs can only go on the inside 2-pole, never the outside pair. Even if the outer ones have handle ties, which some do, those handle ties are merely decoration and do not provide common maintenance shutoff, which MWBCs require. They are labeled as such.  

If you have a GE Qline panel with the cruciform bus bars which allow half width breakers, then you may use the GE half-width 2-pole breakers for an MWBC.  These look like a "duplex/twin/double-stuff" breaker, but are factory handle-tied and also have common trip.  You must never, never use an untied double-stuff.