Are discount furnaces safe to use?
There should be zero difference in the safety between the least and most expensive furnace.  The major differences will feature related.  Any modern furnace purchased in the US meets the appropriate ASTM, NFPA, et al, specifications.

More expensive furnaces will features like:

Humidifers, multi zone heating ability, economizer, multiple heat exchangers for additional burner efficiency, VFDs to vary the blower fan speed, more advanced burner ignition systems, improved filtration and some even come with electrostatic precipitators built in to help improve your IAQ (indoor air quality).  

However, you will not get a "safer" furnace by purchasing a more expensive one.  I guess there is one caveat - eliminating a pilot light ignition is safer as you don't have to worry about the light blowing out and having that gas leaking, but I don't think anyone is building pilot light furnaces anymore (don't quote me on that).  So if there's any recommendation I can make - buy one sans pilot light.

And just like any other consumer product - a lot of times you will pay a big premium to purchase products from a marketing company which has an HVAC division... (if you catch my drift)  

Make sure to get at least 3 estimates to replace that furnace from reputable mechanical contractors (not big box store resellers who just farm out work to a mechanical contractor).