Does it matter which way the air filter is installed? Yes, it does.
Filters are an important part of a forced air heating and cooling system. When missing, particulates in the air (dust, hair, etc.) can build up on the blower and heat exchangers restricting air flow and reducing efficiency. Installed incorrectly, filters can be sucked into the blower mechanism, entangle and jam the blower blades, requiring repair.
At a recent home inspection, we found an air filter installed backwards - the arrow on the filter was facing in the opposite direction of air flow. The arrow on the filter should be installed in the direction of the air flow (from the return ducting to the supply ducting). Upon removing the cover of the unit, the filter was found to be severly buckled towards the suction of the blower. Luckily, the blower was not damaged (yet).
Remember to replace your air filters every three months, and when installing the filter, remember that the filter direction matters - the arrow on the filter should be facing in the direction of the air flow.
In the picture above, 'return' air (red arrows) flows down, through the filter (yellow box), is pulled into the blower and is pushed (blue arrows) back into the living space via the 'supply' ducts.
In the picture above, the filter is distorted, bulging towards the blower on the right side of the picture. The filter needs to be replaced and a new one properly installed.