Yes, it is permissible for a woman to show her eyes. That is so that she will be able to see. But it is not permissible for men to look at the eyes of a woman.
Shaykh 'Abd-Allaah ibn Humayd (may Allaah have mercy on him) said:
If the burqa' [a kind of face veil] covers the entire face, leaving only the eyes uncovered, there is nothing wrong with that. But if it does not cover the entire face, rather it covers the mouth and leaves the rest of the face uncovered, that is not permissible, especially in the presence of non-mahram men.
So the entire face must be covered, but the eyes may be left uncovered so that the woman can see where she is going, as was said by Ibn Mas'ood, 'Ubaydah al-Salmaani and others. And Allaah knows best.
Fataawa al-Mar'ah al-Muslimah, 1/393, 394.
And Shaykh Saalih al-Fawzaan (may Allaah preserve him) said:
There is nothing wrong with covering the face with the niqaab or burqa' [kinds of face veil] which has two openings for the eyes only, because this was known at the time of the Prophet (peace and blessings of Allaah be upon him), because there is a need for that. If nothing shows but the eyes, that is acceptable, especially if that is what women customarily wear in that society.
Fataawa al-Mar'ah al-Muslimah, 1/399
But we should note that the majority of women, nowadays, do not stop at uncovering the eyes only, rather they go beyond that and uncover part of the forehead and nose, so they go beyond the area that it is permitted for them to show. Hence some of the scholars – such as Shaykh Ibn 'Uthaymeen (may Allaah have mercy on him) – forbade wearing the burqa' and niqaab, because of the careless attitude on the part of some women.