Whether it is possible to see the Red Sea from Ethiopia depends on
- Air conditions
- Whether the view is blocked by other high-altitude land or not
For the last two points we have the website HeyWhatsThat, which uses information on topography to generate "views" from mountains all over the world. Browsing the map close to the Ethiopia-Eritrea border shows at least one area of high altitude in an area that Wikipedia spells Dallol and Google Maps spells Dalol. From at least one peak here one would be able to see the Red Sea if atmospheric conditions were right. Sadly, Google Maps does not have a name for this peak nor any nearby place names.
Link to panorama
Screenshot with areas visible from this peak marked in red:
One could play around with the HeyWhatsThat tool to look for other peaks in more accessible parts of Ethiopia with a view of the Red Sea.
Edit: There also appears to be views further south, from where you can also see peaks in Yemen: