What are the disadvantages of more suspension travel?
Although this isn't explicitly part of your question, I'll go ahead and throw it in as it's one of the most important factors to consider in terms of increased suspension if you plan on pedaling your bike- the basic principle behind propelling a bike is to convert a mostly downward force (pedaling) into forward momentum (the drivetrain turning the back wheel which in turn moves you and the bike). The more suspension you add, the more difficult it becomes to transfer the energy of pedal strokes into the drivetrain. The better circles you spin the less you'll suffer from this, but the fact remains that even with clipless pedals the majority of the power input to the drivetrain is coming from your glute, quad, and calf pushing down on the pedal during the powerstroke. Extra suspension soaks this up- it's like running on a trampoline.

Now that that's out of the way, let's talk about how additional suspension affects handling... after we briefly cover the benefits of adding suspension travel. Additional suspension will not only smooth out bigger, abrupt bumps (rocks, roots, your riding buddy that crashed in front of you, etc), but it will also smooth out more gradual terrain changes at higher speeds. Additional suspension basically allows the bike to track better and for the rider to maintain better control of the bike over rougher terrain and at speed. More suspension also helps the bike and the rider absorb bigger drops. Combined with a slacker headtube angle and longer wheelbase (all 3 tend to come as a package) you're also given a larger margin for error; mistakes that may have sent you over the bars on a cross country bike may be recoverable on a bike designed for bigger hits. These benefits come with tradeoffs, however.

Additional suspension will cause "muted" handling- that is that the bike won't feel as sharp or as connected to the trail. It will not turn as quickly or as precisely. This is due to a multitude of factors that come along with longer travel bikes:


Wheelbase is longer because chainstays are longer. This is partially by design and partially because in order to make more room for more travel in the back, the rear wheel typically has to be moved out so it doesn't buzz the seat tube at the bottom of the suspension travel. Longer wheelbases make for bigger turning radiuses. Think of how much room it takes to turn a truck around vs a sports coup. You'll suffer the same sort of effect in tight turns. For a little more detail on how chainstay length affects bike handling, see my answer to another question here.
A slacker headtube angle almost always goes along with more travel. There are exceptions to this rule, but as you move from cross country to trail to all mountain to downhill and freeride bikes, the front of the bike will get more and more raked out. This stabilizes the bike at higher speeds but makes the bike handle "slower" at lower speeds. Slacker headtubes also make for more margin for error in terms of rider input through the bars, but in trade it takes more input to make the bike actually turn.
Weight increases. Bigger hit bikes use bigger tubes to make up the frame, more rugged components, and bigger, heavier duty forks and rear shocks. This is pretty much inevitable as it improves durability and stiffness. That extra weight equates to extra inertia that you have to move around under you, and in general that slows down bike handling.
More travel equals more time to change directions. One of the biggest reasons for this is that it takes a while to load all of that suspension up. You have to push against the shocks hard enough that they start push back harder than the forces of inertia in order to change direction, and this becomes especially noticeable in tight turns on long travel bikes. Other factors play into this, but suspension is the biggest one.
Brake dive goes up as suspension goes up. If you're not familiar with brake dive, it's when the front compresses down (and for full suspension bikes the rear typically comes up), which changes the geometry of the bike. The more suspension, the more dramatic the change in geometry under braking. Modern high end forks have a multitude of adjustments to combat this (slow speed compression adjustment, threshold adjustment, etc) but there's currently no technology that will fully make up for this, and the longer the travel in the front, the worse this will be.
As travel increases so does bottom bracket height which, as Vincent pointed out, raises the bike's center of gravity. Obviously this isn't a 1:1 ratio in terms of additional suspension and additional bottom bracket height, but it is pretty much unavoidable and does have an effect. As is the case with most of these points, there is typically some form of compensation for this. Some trail and all-mountain bikes come equipped with dropper seatposts (hit a lever on the bar and the seapost telescopes down) and DH/FR bikes have their seat height set much lower than a standard XC bike. Dropping seat height allows the you to drop your center of gravity, but ultimately your feet are on the pedals and that height is set, so the you will start out higher off the ground on longer travel bikes.


I'm sure you've noticed that I've talked a lot about other factors beyond purely the travel of a bike's suspension alone, but it is important to take into consideration the whole package as all of the factors mentioned here work in symphony to create the ride quality of a bike as a whole, and each factor attributes to the ride quality in its own way- both positive and negative. You can't consider suspension travel alone because you can't find a bike where all else is equal beyond travel. You can't take a cross country bike and give it 8 inches of travel and call it a downhill bike anymore than you can take an F1 car and add 3 feet of suspension to it and call it a Baja truck- Each is a purpose built platform in nearly every aspect.