Is RNase AWAY in the lab dangerous?
RNase AWAY is marked for Category II skin corrosion/irritation and Category II eye irritation/serious eye damage. Recommendations by the MSDS followed by definitions as follows:

Skin contact Wash off immediately with plenty of water for at least 15 minutes. Remove and wash contaminated clothing before re-use. Immediate medical attention is required.
Eye contact Rinse immediately with plenty of water, also under the eyelids, for at least 15
minutes. Immediate medical attention is required.

SOURCE
DEFINITIONS FOR GHS LABELING
More information is included in the MSDS provided.
As for any smell, RNase away is reported to be a proprietary alkali hydroxide solution, so without any additional information it's difficult to tell.