Yes, there will be problems, because Nanodrop simply measures a spectrum of the optical density, what can inform about purity and concentration of "nucleic acid", but not about its quality.
I suggest you should keep some gap between both the measurements. Cleaning the pedestals and calibration is very important. or else for a foolproof estimation use a fluorometer (Qubit, Invitrogen) which will give you the most accurate information about your concentrations.
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I don't see why that would be a problem, as long as you clean it properly and choose the appropriate nucleic acid on the display. If you're really concerned about it, I believe you can clean the pedestals with HCl 0.5M, or a manufacturer's cleaning product; but I wouldn't clean that way very often. Just dH2O seems to work fine, in my opinion. Something I always do is to measure a blank (dH2O) between each sample to wash the sensor.
Also, Nanodrop has its limitations. It will give you very accurate 260/230 and 260/280 ratios (sample purity); not so accurate sample concentration readings; and won't give you any information about sample integrity. For me, those three are the most important quality indicators for DNA/RNA samples.
For a real concentration measure, I would recommend Qubit. And for sample integrity, you could use Bioanalyzer (RIN reading), or just run it on an agarose gel (cheap and fast, and can help you see differences in concentration, too!).
I am confiused by the answers given. The question was about quality. I don't see how these instruments (Nanodrop, Qubit) can determine the quality of nucleic acids. They can measure the concentration, the spectrum of the Nanodrop also allows to judge the purity. The quality is usually related to the fragment size distribution (integrity), less commonly by the amount of modifications (adducts, base losses etc.). To get the fragment size distribution one needs some kind of electrophoresis (PAGE, agarose gel, Bioanalyzer, Tapestation, etc.). Modifications can only be analyzed with more complicated biochemical or biophysical assays.
Jochen Wilhelm while I partially agree on your definition of quality, I think the question was if there is a problem using the same Nanodrop device to measure both DNA and RNA, or if that would affect the Nanodrop readings.
You cannot measure quality (meaning molecule fragmentation/degradation vs. intact) on a Nanodrop. Yes, you can use the same machine for DNA and RNA. You just have to clean off the pedestal really well between samples.