According to the table for fire forensics* (reference below), plastics melt in the temperature range of a low-pressure steam heating pipe. From the online / book research I've done pipes and radiators max out at around 120°C, and are typically closer to boiling point (100°C). **
Forced hot water operate at lower range, typically 60°C-82°C (140°F-180°F) depending on the load on the system. If your needs do not exceed operating at 140°F (60°C) regular network cable might be fine. There is also industrial high temperature cat5 cable that operate at 75°C even 80°C. I consulted globalspec.com and cable data sheets. From what I've found there is not an easy universal way to find the operational temperature from looking at the cable. You might be able to use the UL listing number or standards codes if these are printed on your cable, or information on the packaging or receipt to find the data sheet for the cable you have.
You could insulate your cable, but insulating the heating pipe will offer more protection and also save energy.
According the same fire forensics table the lowest ignition temperature for plastics is 349°C, far above any of these operational temperatures for water or steam heating systems so there is no fire hazard whatsoever from running the network cable near heating pipes. Fabrics have a much lower ignition temperature, and even these temperatures are far above the operational temperatures for steam and forced hot water home heating systems.
Electric baseboards are another matter. Do not network too close to them. Also, don't network next to your dryer duct or for that matter your boiler flu pipe. Dryer exhaust pipes, especially from a gas heated dryer, can be very hot - much hotter than pipes for forced water or steam heat. Blocked dryer vents are one of the most common causes of house fires, so clean yours out regularly.
While I believe this all to be factual and sensible, I am a software developer, not a fire marshal or building inspector. This should not be taken as professional advise and comes with no guaranty what so ever. Proceed at your own risk.
*Table data source: http://www.tcforensic.com.au/docs/article10.html#2.1
Operational temperatures for heating systems were taken from a variety of sites about heating system maintenance and fire safety. Please verify the numbers for your system by taking some measurements.
** High-pressure steam systems in industrial, and commercial settings operate at considerably higher temperatures than home heating systems.