since DNA loading dye for agarose gelelektrophoresis generally consists just of a tracking dye and glycerol/sucrose I can't imagine any significant interaction with the DNA. The purpose of gylcerol is to increase density of your probe so it doesn't float away while pipetting it and that it only starts traveling through your gele when you apply current. Unlike ununiformly charged proteins which would be significantly altered in their travelling speed, DNA doesn't need to be eqaulized chargewise through detergents like SDS.
I do not think that glycerol interacts with dna but you may see a small mobility ( therefore size) effect if your loading dye is made up in a different buffer than your gel, size standard or dna sample. For instance if you added a large amount of loading dye in a high salt buffer then the current in that track would be different from the size standard track so the 2 samples would run at slightly different speeds so the size of your sample dna would appear to change...too much salt = lower current = slower running = larger size dna