Firstly, lasers of ANY wavelength/colour can potentially cause eye damage if their power exceeds 5mW, and wavelengths less than 400nm or greater than 1200nm WILL still damage eyes.
In the case of wavelengths less than 400nm the damage will be considerably greater than that from a visible laser, as this is the ultraviolet portion of the spectrum (the same light which causes skin cancer from over-exposure to sunlight).
Light over around 900nm is in the infrared region of the spectrum, and while invisible, definitely CAN cause damage to the eyes and again, as with UV is considerably more dangerous that visible light. As you can't see infrared light, your blink reflex won't help minimise exposure if the beam enters your eye. Rather than seeing a bright flash, blinking, and turning away, thus minimising retinal damage, the first sign that you've messed up will be your vision simply disappearing in the parts of your eye the laser hits, as your retina is burned, perhaps irreparably.
What you were likely thinking of in regards to the eye "filtering", infrared light was the fact that, to long wave infrared light (around 10um which is equal to 10000nm), many materials which are transparent to visible light, like glass and the lens of your eye, are no longer transparent. Rather than passing through these materials, the light is absorbed. This means that rather than instantly punching a hole in your retina and blinding you, such lasers would take some time, as before reaching your retina, they'd need to vaporize the lens of your eye (so maybe not INSTANT blindness, but searing pain followed by eventual blindness). Light in this area of the spectrum is only really available through the use of carbon dioxide gas lasers (10.6um). A LOW powered CO2 laser easily puts out 20-40W, enough to set virtually anything on fire, and cause a glowing white hot spot on any surface the light is allowed to fall on. On top of that, the lasers are large and heavy and run off around 18,000V DC, not something I'd recommend.
Any UV laser small enough to be used in such a show would be well out of your price range($1000s), not to mention the fluorescence UV light causes in materials. Have you ever seen white materials glow under a "black-light"? That's an example of what happens when low energy low intensity UV interacts with a fluorescent surface. High energy high intensity UV like that of a laser would cause FAR more fluorescence, so even though the light itself is invisible, you could still see where and what the laser was doing.
The only somewhat commonly available UV laser is a nitrogen gas laser, which emits 337.1nm, but even then, it's a pulsed laser and has average outputs too low to do what you describe.
High power infrared lasers are more commonly available, have a read about 808nm diode bars (slightly visible to some people), and pulsed solid state Nd:YAG lasers if you're interested, but again these can all cause severe and permanent eye damage.
200mW can pop a balloon at close range if held still for a few seconds, to easily pop a balloon unfocused at several metres without a tripod you'd need considerably more power, which I wouldn't recommend, as even 200mW can cause permanent eye damage.
The only way this would be even remotely safe was if every person in the audience had a good quality pair of laser safety glasses rated for the wavelength being used.
I'm sorry if I sound a little harsh, I'm not trying to dissuade you just for the sake of it, but with this kind of thing there's a very real chance of serious eye damage.
Go with 405nm, it's great for burning. Also if you might want to also try gas lasers. They are certainly invisible.
405nm isn't invisible, it's violet, and only some gas lasers (Nitrogen CO2), are invisible, Argon, Helium-Neon, Argon/Krypton Copper Vapour etc. are all visible, as are many more.
