The components used to get dry air are:
Metal distribution pipes with drain valves
Water separator/filters
pressure regulators
Coalescing filters
Desiccant Filters
TP tools has a page with the basic plumbing of an air line system
Air Line filters
The Water separator/filter will filter out large particles and droplets of water. The filter separator will have a bowl to catch the captured water and a drain to remove accumulated water.
A regulator placed in line will reduce the pressure and temperature of the compressed air forcing more condensation of water vapor.
Coalescing Filters have an element which looks a lot like a roll of toilet paper. The compressed air is forced to follow a serpentine path through the filter. This brings the small droplets of water together forming droplets large enough to stick to the filter. After use for 'a while' the filter element in the Coalescing filter will be wet and must be replaced. (The wet filter can be dried and reused.)
<A target="_blank" HREF=http://www.tptools.com/prod_list_display.asp?dept%5Fid=L2%7E73&dept%5Fname%5Fp=Paint+Sprayers+%26+HVLP&mscssid=KUM3H57XCFTU9KFALFJDX44XMAR34R49>Desiccant Filters</A> run the compressed air through a bed of a desiccant material like silica gel which can remove water vapor. When the Desiccant bed is saturated with water vapor the bed must be replaced or baked in an oven to remove the vapor and returned to the bed. Desiccant filters are used for sensitive spray painting. When you are spraying $70 per gallon paint and a little water vapor will ruin a paint job they are useful. For less critical applications they are usually not worth the expense.
A package with a water separator, pressure regulator and coalescing filter after metal distribution pipe is good enough for all but the most critical purposes (see <A target="_blank" HREF=http://www.tptools.com/product.asp?base%5Fno=3411%2D06&str%5Fbase%5Fno=3402%2D00%2C3403%2D00%2C3405%2D60%2C3409%2D00%2C3411%2D02%2C3411%2D03%2C3411%2D12%2C3411%2D06%2C340%2DGAUGES%2C3500%2D60%2C3407%2D00%2C3405%2D60B%2C&header%5Ftitle=Air+Line+Hookup%2D1%2F2"+Inline+Water+Separators%2C+Air+Regulators&page%5Fname=prod%5Flist%5Fdisplay%2Easp&search%5Ftype=L2%7E113&size1=&size2=&gender=&ShowImages=yes&sq=0&cont=1&intPgNo=1&mscssid=KUM3H57XCFTU9KFALFJDX44XMAR34R49>Filter Regulator System</A> (I just purchased this package to filter the compressed air going to my plasma cutter.)
Are you sure you are connecting the filter up in the correct direction? Connecting the filter outlet to the compressor can easily blow out filter elements.
Does your 'tire inflator' have an air tank? If not and if you are running the motor continuously for longer than the time to blow up a tire, you will have problems. I would expect a very short life for any compressor running continuously. The rule of thumb is that the compressor should cycle on and off filling the air tank, being on for 1/2 to 2/3 of the time for heavy duty and commercial quality equipment. For a low duty cycle device like a tire inflator I would expect a short life with the pump running for more than a few minutes per hour. In addition to fabricating an accumulator(tank) (as others have already mentioned) you will need a pressure switch to cycle the tire inflator on and off to maintain the pressure in the accumulator within limits and avoid running the compressor continuously.