Strive to enter through the narrow gate, for many, I tell you, will attempt to enter but will not be strong enough.
After the master of the house has arisen and locked the door, then will you stand outside knocking and saying, 'Lord, open the door for us.' He will say to you in reply, 'I do not know where you are from.' And you will say, 'We ate and drank in your company and you taught in our streets.' Then he will say to you, 'I do not know where (you) are from. Depart from me, all you evildoers!'
And there will be wailing and grinding of teeth when you see Abraham, Isaac, and Jacob and all the prophets in the kingdom of God and you yourselves cast out.
This and other passages pretty much rule out asserting universal salvation, i.e., that everybody will make it to heaven. To be sure, it is not formally heretical to allow for the possibility of universal salvation; in a relatively trivial sense, that possibility holds; for its holding is logically compatible with what has been irreformably taught by the Church. Since Christ's Passion is sufficient for all, everybody can make it to heaven. But I don't think Scripture and Tradition are hospitable to the idea that everybody actually does make it to heaven. It seems much more likely, given Jesus' language in particular, that some people will be damned. (For a good overview of this topic, see Cardinal Dulles' article "The Population of Hell.") But what's most interesting to me is the sub-population the Holy Spirit seems to have in mind in the present passage.
On an obvious, historical level, Jesus seems to be speaking about certain Jews of his day. Assuming he himself is the "master of the house," he has himself saying someday to certain people in the parable "I don't know where you come from," banishing them permanently from the household of God they want to enter. Not a few Jews of Jesus' day rejected him despite having eaten and drunk in his company and heard him teach in the streets. And Jesus duly warns them. But it's never that simple. The Lord's words apply as much to the future, to his Church, the new Israel, as to those present when he walked the earth. It is quite conceivable that some who are baptized, who are raised in the true Faith, who go to church and take communion often, will find themselves being told at the end "I don't know where you're from. Depart, evildoers!" Some people who are formally in the Church, the household of God, are not followers of Christ in their hearts, despite claiming to be and having a velleity, as distinct from a will, to do so. And they show that by how they live.
I think that phenonomenon is at least as common today as it's ever been. Legion is the number of people who claim to be Christian, of whatever variety, but who live lives indistinguishable from that of your average unbeliever. Legion are the Catholics who claim to believe what the Church teaches, and might actually do so up to a point, but only so long as it doesn't cost them much beyond a fin in the collection basket. For such Catholics, religion only goes so far; it becomes a different matter when following the Church's teachings on, say, social justice and/or birth control will cost us dearly. Then we come up with all the arguments we can—if we can be bothered with arguments—that such teachings are only matters of opinion, and therefore optional, and therefore safely ignored. Keeping religion out of the boardroom and the bedroom, after all, is so much more mature and realistic; it would be irresponsible, if not actually sick, to get too rigid about these things; and so on. We've all heard it before.
Such are the lives and attitudes of the people I think Jesus is advising to "strive to enter by the narrow gate." Actually, none of us are strong enough to do so; we should take that as a given. But if we don't even strive to do so, under the power of grace, then the grace that is always on offer won't "take." What's sufficient objectively for salvation, namely, what God has done for us in Jesus Christ, will not be enough subjectively. Many so-called believers fail to observe that distinction. When the failure is culpable, they cross the line between the assurance of salvation and the sin of presumption.
Those who sincerely put their faith and trust in Jesus Christ will indeed be saved. But there's a much misunderstanding of what that kind of assurance means. Among both Protestants and Catholics (I cannot speak knowledgeably about Orthodox), it is very common to assume that if you more-or-less believe and are a more-or-less tolerable human being, you will be saved in the end. As a result, many of us have little spiritual life to speak of; the effort and the self-abnegation hardly seem worth it if, without them, you can still make it through that door before the Master closes it. Yet Jesus' words supply no justification for that approach to life. The question we really need to ask ourselves is not so much where we are as what direction we're headed in. "Many of the first shall be last, and many of the last shall be first." If we think where we are is somehow good enough, then we're not headed in the right direction. And the longer we postpone a change in direction, the harder it gets to make the change. Some of us will probably find, in the end, that we have presumed too much for too long, and that it is now too late.