I ran into that issue myself. I submitted a compound for NMR in DMSO-D6 and needed the compound back. Essential oils are usually somewhat volatile.
There are two things to try:
If your essential oil isn't soluble in water, just extract the oil using water, and your compound dissolved in dichloromethane, ethyl acetate, hexanes, depending on which organic solvent dissolves your compound. A couple of water washes will remove the DMSO. This is probably the easiest method.
Use a small C18 column, essentially solid phase extraction. One can use a small open column, but I used a small flash column. Details here: http://www.isco.com/WebProductFiles/Applications/101/Application_Notes/AN97_Removal%20of%20Non%20volatile%20Solvents.pdf If your compound(s) don't elute in methanol, the column can be washed with methanol, and the compounds eluted with ethyl acetate or dichloromethane if using silica-based C18. I find this method works better for more polar compounds that are water soluble, yet retain on C18.
The suggested procedures by Dr Silver and Dr Devangi may be work well but, normally, essential oils are complex mixtures containing also partially water soluble components or with some polarity which cannot be extracted efficiently or may be retained in column during chromatography. I have some doubt that you may obtain an oil with the same original composition by extraction.
Another procedure to eliminate DMSO is reverse osmosis but I'm not sure that it may be easily applied in this case. In any cases you may lost a part of your sample. And also the amount of the sample may have a role in the choose of the procedure to apply.
Let us wait for other contributions to this topic.