Since the question does not specify a denomination, I am assuming that any denomination is OK. I thus am basing my answers with one of the clearest doctrinal statements I could find. However, it is pretty representative of most arguments that argue against obligatory head coverings, coming from a variety of theologies.
Here is a statement by the Reformed Presbytery In North America, which I believe is considered pretty conservative on most issues. The argument is that the context of 1 Corinthians shows it is an example of Paul's larger argument of being culturally sensitive, not an instruction for all people in all cultures. In the later part of chapter 10, Paul discusses meat offered to false idols. He says that it would be unlawful for a Christian to eat such meat in the context, because that would constitute worshiping the idol.
10:21 You cannot drink the cup of the Lord and the cup of demons. You cannot partake of the table of the Lord and the table of demons. (NRSV)
Immediately after that statement, however, he says
23 “All things are lawful,” but not all things are beneficial.
(The quotes are added by NRSV because it is seen as something Corinithians were saying to justify their behavior. Paul responds not be denying the statement, but by saying there are other considerations in play than just lawfulness.) Paul goes on to make it clear that it is not the act of offering it to the idol that makes the meat unlawful, but the cultural context. He goes on to say
27 If an unbeliever invites you to a meal and you are disposed to go, eat whatever is set before you without raising any question on the ground of conscience. 28 But if someone says to you, “This has been offered in sacrifice,” then do not eat it, out of consideration for the one who informed you, and for the sake of conscience— 29 I mean the other’s conscience, not your own. For why should my liberty be subject to the judgment of someone else’s conscience?
Thus he is saying 27) don't cause offense by asking if the meat had been offered to an idol, but also 28-29) if you know it is, don't eat it and appear a hypocrite, because (implied) this may inhibit the unbeliever from being Christian.
Next comes the head covering passage at the start of chapter 11 and then instructions about receiving the Lord's Supper. He says that some member have been causing offense by making their higher social status known during the meal. He advises these members to "eat at home" so as to not cause offense by displaying their cultural status. Or as the linked document says:
similar to Paul's concern for godly order, decorum, and the eschewing of offense in 1 Corinthians 10 (in regard to meat offered to idols), Paul instructs the Corinthians how they should order the circumstances that surround the celebration of the Lord's Supper. He teaches them that it is offensive and divisive to fail to wait for one another, and that if the reason one cannot wait is hunger, then it would be expedient to eat something at home before coming. Although, we may never see this particular offense arise in our circumstances, nevertheless the principle of unoffensive behavior in a public setting is applicable to many circumstances.
The argument continues that the next several chapters cover similar themes - the orderly use of spiritual gifts and how to conduct church service in an orderly fashion - which can be understood as respecting cultural norms. Thus
In keeping with this context, we believe that Paul in 1 Corinthians 11:2-16 was continuing in the same general line of argument. He addresses the headcovering practice which was culturally acceptable to the Corinthians, and seeks to teach them that they are not to be offensive, divisive or contentious by altering the customs of the land, when they come to worship. He is laying down the same principle as that taught both before and after the headcovering passage. In effect he is saying, do not alter the established order of this circumstance when you see that it will be offensive and destructive to the unity of the church.
Since it is no longer a cultural norm to where head coverings for women, the advice no longer applies.
To cross-reference, here is another argument by a believer coming from a completely different background. He adds some points (such as the setting of Paul's instruction likely not being a church service) and explains the cultural background, but the basic argument is the same - the context of the passage shows it is not a specific instruction, but rather part of a general argument to respect cultural sensitivities. I imagine this would be the route taken by most who would not adhere to a "headcoverings are required" doctrine. This reasoning wouldn't apply to (most) other passages, because there is no surrounding context saying "X is lawful, but should be avoided anyway" in other NT writings.