I often approach deer to see how near I can get and to watch them (just for fun - not to hunt). I take care for the wind and of course to be silent enough.
What else should one consider stalking/approaching deer?
The biggest things are scent, movement and not seeing what the deer can see. Most people trying to approach a deer are not seeing the forest, field, meadow or whatever, as the deer sees it. Crouch down so your head is about 2-3 feet lower than normal, and the world looks different. You see different paths through the brush, and a lot of times you discover that you thought you were hidden, but you were exposed and there's almost a viewing portal between you and the deer. One of the most enlightening experiences I had was pushing into a thicket up in the mountains that was so many acres in size, so thickly overgrown, that I had to get on my hands and knees and crawl for an hour on game trails to get through it. The world looks different on deer level.
Other times, just movement and projecting a calmness. I've sat on a log after coming back from a run, in the snow, just sat down and stayed still, out in the open and had 7-8 deer come out and feed up right to within about 15 feet. They all saw me, knew I was there, but I just sat there smiling and still, and enjoying the moment. Was hard to do, I had decided I was going to run bare chested, so I sat there in the snow with no shirt on, freezing my rear off. Maybe they figured I was freezing to death and was no threat? But I believe animals can sense intentions as well as detect them (via pheromones as much as movement, speed, etc).
Practice catching armadillos and sneaking up on squirrels, you'll learn a lot. As a teenager, my buddies and I would see who could snatch the most armadillos up by their tail in open fields at night. They have ridiculously good hearing, but poor eyesight. You learn how to sneak having to constantly reposition yourself to get in their path, as they rarely root around in a straight line. Then, you have to either at the last moment find a way to get behind them or let them walk almost right into your feet. When you or they get close enough, you reach down and grab them by the tail and snatch them up (be sure to hold them stiff armed, at arms length, away from your body...those claws allow them to burrow into the ground in seconds for a reason). And sneaking up on squirrels...hard to do, but it's entertaining to see their reaction.
On the movement, timing is important. Not just moving when something isn't looking your way, but your movement pattern. Don't take two steps or four. Take one or three, and then a pause. Four legged critters don't move in the same sequence as us bipedal types. Even if they can't see you, if you're close, they probably hear you. They can be looking away from you and know you're a threat because they heard a movement that isn't a norm in the woods. And off they go.
A lot of times, if you jump up a bedded buck he jumps up and looks around. If you had the wind in your favor, maybe he's heard your movement, but doesn't know where you are exactly. Even if he doesn't see you, instinct tells him to take off because he heard a wrongness in the movement he heard, so off he bounds. Then, sometimes, if you snapped a twig or made some obvious or even loud noise and jump one up, they might look around and then sort of quickly walk off as opposed to running/bounding. He's probably decided that the noise startled him, but he hasn't seen you, and the noise wasn't 'wrong' but it's best to go anyway, because you never know. That's how bucks grow to be big bucks.
The animal may startle causing harm to itself or others.
For example Nature Workshops says:
Wild animals that have been approached too closely, have:
Been startled and run into traffic where they have been hit by vehicles.
Lost footing on cliffs and fallen.
Fled suddenly and suffered injury or death while trying to escape.
Been separated from their young or abandoned their nest.
Been distracted from watching for predators.
Acquired a human scent track that can lure predators to nests or young.
Abandoned an important food source, reducing their chances for survival.
Responded aggressively without warning, posing a danger to those watching them.
Only aggression poses a direct threat to you, but all of them are bad for the animal.
There are three things deer will pick up on
Another item to be aware of is that they will also react to a sudden change of behavior. I've been walking noisily in the woods with friends and passed within 20' of deer that didn't care at all until we changed behavior (stopped talked, stopped walking, etc.)
When you are approaching, try to only move when their heads are down. Watch for tail swishing as they do this more when agitated. If they are agitated, just chill and be still for a while.
As pointed out by James, large herbivores are not harmless. They may run away when you surprise them (which you eventually will) or they may decide to trample and gore you. The average person is no match for even a smallish buck.
You should consider that an angry or surprised deer can decide to attack. Deer kill people, excluding the 120 to 200 or so by auto accident in the US, deer also kill by attacking. reference1 reference2 I was not able to find accurate numbers but in the US count seems to be less then 52 per year, probably in the 1 to 5 range.