In my experiences, dsDNA is very stable in room temperature and I was able to get a DNA without problems after about a month at room temperature. However, I recommend you keep DNA on fridge for your long-term uses.
DNA is unstable in pure water containing no counterions in excess i.e beyond 1:1 stoichiometry (in case of monovalent metal ions) to the phosphates present.Of course when you dissolve lyophilized DNA you also dissolve salt impurities present and therefore you may get stabilization at room temperature for long times.
It actually depends on the extraction method as well. For example, if you do Miller et al., 1984 - salting-out method, which is free of toxic chemicals, you will get DNA of high quality. Now, if you leave it on the bench and forget about it for a week or so, you will find an abundant microflora feeding on your DNA. If your starting material is nicely preserved tissue than a couple of days room temperature will not harm it. However, I wouldn't keep it at +4 C longer than 2 weeks either.
I think its depends on the extraction method, and final elution material for example the elution with TE buffer maintain DNA for long time rather than aqueous solution. further more the quality of DNA and presence nuclease effect also should be taken in consideration.
Pure DNA is unstable in pure water .The reason is simply that the DNA Duplex is a polyelectrolyte and falls apart into single strands if the electrostatic phosphate-phosphate repulsions are not screened by salts,aggregation or charge neutralization via ligand binding (e.g. multivalent cations,nucleoproteins,spermine etc etc).How long it can survive as an intact duplex depends on all components present in the actual solution,the pH,temperature,pressure and DNA concentration and DNA length.For instance,I have lyophilized DNA (i.e.solid state fiber aggregates with all their counterions sitting on the phosphates ) in my refrigerator for 30 years now but if a dissolve it in pure water and low DNA concentration it will become a single strands soup
in at most few days.This so called dilution denaturation has been experimentally studied by early DNA pioneers (e.g.Inman,Michelson) and it is also clearly seen in honest computer simulations not cheating with the electrostatics (e.g setting the phosphate charges equal to zero,or setting the coulomb interaction equal to zero for distances greater than 12 Angstrom,etc,etc.),where the duplex starts to fall apart after a few picoseconds)