Is invasion of privacy with no monetary loss illegal according to halacha (Jewish Law)? For example, using another person's phone without permission, or even touching it without actively using it? (ie. looking at the screen)
To the Questioner,
To move and touch someone else's phone or computer without permission is prohibited, according to the rule we have of “One who borrows without permission – is considred as stealing”.
Source: Shach, Choshen Mishpat, 358:1.
To look at an open screen depends on the situation. If the person using the device left it open for everyone to see, then there is no prohibition of invading privacy.
If, however, it is necessary to uncover something to see the screen, for example if a cellphone was placed upside down or in a drawer in a way which shows that the owner did not leave it out for others to see, it is logical that this is included in the Cherem of Rabbenu Gershom who prohibted reading another's personal letters.
Most certainly if one has access, for whatever reason, to the password of another's email account, he cannot enter the email without permission of the owner as it is included in the Cherem of Rabbenu Gershom.
Soure: Be'er Hagolah Yoreh Deyah, 334.