Does sexual orientation influence gender bias? Let's take three people: Alice, Bob, and Carol. Suppose Alice is heterosexual. That should imply that Alice finds Bob attractive at some level, but not Carol. Does this mean Alice is more likely to be biased in favor of Bob (and the reverse if Alice is homosexual)?

I've not found anything directly analyzing this. Closest was this paper, The Influence of Sexual Orientation on the Perceived Fit of Male Applicants for Both Male- and Female-Typed Jobs, but it dealt with whether the sexual orientation of the applicant affected whether the applicant was perceived by the hiring manager as suitable, i.e. it's not the sexual orientation of the hiring manager that's being investigated.

Presumably if the answer to the title question is "yes", then it would be advisable to appoint asexual people to e.g. settle sexual harassment complaints.
The issue around attraction needs to be clarified and assumptions challenged and proven. 

To imply that sexual orientation is solely and wholly responsible for attraction is a severe and gross oversimplification verging on stereotyping and discrimination. As Wiki argues physical attractiveness is distinct from sexual attraction. Just because someone is physically attractive and also falls within one's sexual orientation, that person may also not be attracted to the person. The person may already be in a fulfilling relationship or the other person has a wedding ring, to imply that humans are animalistic and therefore their desires translate into behaviour without much other consideration is quite improper and unscientific. You should not draw a specific scenario like studies on hiring interviews to imply that the evidence is not present. There are enough studies on attraction and behaviour in other fields to disprove the specific example of hiring interviewers. I am sure you would not like your race, age or any other stereotyping physical attribute to be the sole determinant for how people and the public assess you? Would you? 

On that note, there has been research on attractiveness according to sexual orientation. If attraction is the basis of hiring decision-making process, this study disproves this. However, if you are implying sexual orientation goes through a different mechanism to affect the decision-making process at the hiring interview, then I would be keen to hear it. This 1992 study of 52 subjects (four groups of 13) found minimal or no difference between heterosexuals and homosexuals on the assessment of attractiveness. Both heterosexual and homosexual men rated younger sex subjects higher than older subjects. Your question implies that homosexual men should rate women lower than men as they are not sexually attracted to them. If there are no differences between attractiveness depending on sexual orientation, it seems hard to argue that asexual people should be used on hiring interview panels. 

References: 

Jankowiak, W.R., Hill, E.M., Donovan, J.M., (1992). The effects of sex and sexual orientation on attractiveness judgments: An evolutionary interpretation. Ethology and Sociobiology. Volume 13, Issue 2, Pages 73-85.