I've experienced the following issues:
Obstruction to the line of sight to the satellite
Anything that gets in between the VSAT dish and the satellite can weaken or obliterate the signal. Obviously when you initially set up a VSAT dish you find the clearest possible view of the sky, but environments have a tendancy to change. Buildings are erected, or extra floors are built. Trees grow taller or grow extra foliage. I've experienced this with mango trees in a couple of locations. Mango trees seem to be common in locations where VSAT is the only connection option!
VSAT operates in several frequency ranges, the most common being:
Different things can create interference in each of these bands. Notably, WiMax operates in the 2 - 11 GHz, which overlaps C Band. C Band installations within 50 miles of a WiMax network can potentially be effected. Nice one IEEE! I have also observed local welding work to completely obliterate a Ku band signal. When the work was completed the connection came back.
Ku band is notorious for being affected by water droplets in the atmosphere - commonly referred to as rain fade - although this can affect any frequency above 11 GHz. This is now usually mitigated against by changes to the transmission power at the satellite and earth stations, but can still sometimes cause problems. Build up of moisture in the feed horn of a VSAT dish can degrade the signal enough to break the connection - if the moisture is removed, or once it evaporates, the connection is reestablished.
Pointing a dish at a small object 35,786 km away is a fine art. Moving the dish by just 1 degree results in pointing at something more than 600 km away from the satellite! Small knocks to the dish can result in loss of signal, but the dish still looks like it is pointing in the right direction. That is why you spend a lot of money and effort on building secure mounts for your dishes, and put up signs suggesting that getting near the dish is dangerous. Leaving dead birds lying around the dish can be helpful! Talking of birds, they have been known to build nests in VSAT dishes - they generally come with some spikes to discourage this. I've seen a 1.5m Maribou stork perching on a VSAT dish - something that can easily change the alignment of the dish if the mount is not strong.
Cabling can also get damaged (and is expensive and awkward to replace). Again, investment in proper cable runs is worthwhile. Wind blowing a loose cable against a wall, especially over an edge, can result in damage to the cable over time.
Like any metal outdoor object pointed at the sky, lightning strikes are a risk. Generally you won't be supplied VSAT equipment without some form of lightning protection, but adequate earthing/grounding is usually left up to you. Are lightning rods a good idea? I'm not sure, and haven't been involved in any installations that include them.
Solar flares and storms can interfere with any radio communication here on earth, and satellite communications are no exception. Twice a year the satellite will pass directly in between the sun and the earth station - a bit like a solar eclipse, except the satellite is not big enough to blot out the sun. This results in potential disruption four times a year (twice a year for each earth station). Most VSAT suppliers will send you regular updates of when these are likely to affect you. The duration is usually around 5 minutes.