Is one allowed to walk by an automatic door on Shabbos, if he/she has no benefit of the door opening (i.e. not walking in)?
How does this case compare/contrast to the walking by an automatic light on Shabbos?
In order to answer this question, we must first introduce a few concepts (which some of the other answers mention as well):
Literally, 'cutting the head' based on the classical example of this case brought in the Gemura in which an individual wants to use the head of a chicken as a toy for his child, but the head is attached to the chicken's body, he does not want to or intend for the chicken to die as a result of his removing the head, but it is inevitable.
Psik Reisha refers to any case in which an individual does an action which is permissible on Shabbat or Yom Tov (ie. separating the chicken's head from its body) but will inevitably cause another result which is prohibited on Shabbat or Yom Tov (ie. the chicken dying).
One who does an action categorized as a Psik Reisha is fully liable for the prohibition he/she causes.
Psik Reisha D'Lo Nicha Ley
Psik Reisha D'Lo Nicha Ley literally means 'a situation of Psik Reisha which is not desirous to him' and refers to a situation in which a person does an action which is categorized as a Psik Reisha, but the secondary action (the prohibited action) is not something which he wants or needs to happen. For example, it if forbidden to water plants on Shabbat or Yom Tov, so washing ones hands over his personal lawn would be forbidden because it is a Psik Reisha, and washing ones hands over a stranger's lawn would be a Psik Reisha D'Lo Nicha Ley.
Generally, a Psik Reisha D'Lo Nicha Ley is forbidden just a regular Psik Reisha.
Melacha Deorita vs. Derabanan
All prohibited acts on Shabbat and Yom Tov are divided into two categories, those things which are forbidden by Torah Law (Melacha Deorita) and those which are forbidden by Rabbinic Law (Melacha Derabanan). Both are equally prohibited to do on Shabbat or Yom Tov (except in extenuating circumstances such as danger or a sick person, but those laws are complex).
Bringing It All Together
Now, although individually all of the above listed categories are prohibited on Shabbat or Yom Tov, when they overlap under very specific parameters they may allow for leniency in certain situations. An examples of such an overlap is as follows:
According to some opinions (I believe Sefardim, based on Hacham Ovadia hold this way, but Ashkenazim, based on the Mishna Berurah do not. Please check with your local Halachic authority.) hold that if you construct a situation in which the action you are doing is a Psik Reisha D'Lo Nicha Ley and the resulting prohibited act is a Melacha Derabanan, you are permitted to do that action.
Therefore, your two cases fall into two separate categories:
I hope this answer helped clarify some of the complex issues involved with the situations you raised. It is not meant as a practical halachic answer, you should seek out a local competent Halchic authority for this purpose.
I think it would be permitted (according to many poskim) if there isn't other way to get to where you are going. It is probably a more lenient case than causing a light to go on. If the light bulb gets hot, there are clear halachik issues involved in turning it on. On the other hand, causing doors to open is not as clearly prohibited in the first place.
no. this would be pasik resha.
the difference to a light bulb is that lighting it is mideoraita. stam the door is mahloket if mideoraita or miderabanan.
also the door there's more room to say that you don't have benefit from it opening.
also note that there's a difference of opinion between maran and rama if pasik resha that you don't benefit is permitted or not.
I can answer the first part, Rav Yishak Yosef (as well as Rav Aharon Zakai) writes that it is asur to walk by a door like this (and he also writes that walking by the place where the light will turn on is Asur).