I believe in biblical separation, but I don't think that is what Amos 3:3 is talking about.
One way of looking at Amos 3:3-8 is that these are examples of "cause and effect." Two men do not go for a walk together unless they have made an appointment to do so. Lions do not roar without some reason, birds are not caught in a snare unless the snare is there, watchmen do not blow the trumpet unless there is some threat (in this case the threat of judgment). In this case, the "cause" is the sin of Israel and the "effect" is God's impending judgment.
"It is not likely that two men will meet in a trackless desert. If they do walk together, it may be safely assumed that this is so because of a prior appointment. . . . The inference is that Amos' voice is a sign that Jehovah's judgment has come." (New Bible Commentary)
Another interpretation is that Amos is saying that God will no longer walk with Israel because they are no longer in agreement.
"It could not be they should long have God's presence with them, or that he should walk among them and bless them, while they walk so contrary to him; they could not in reason hope that there should be any friendly commerce where was so little agreement and friendship; a retaliation they must expect from the Lord; he will forsake them who have forsaken him." - Matthew Poole
Yet another interpretation is that the two who walk together here are God's prophets who agree that judgment is coming upon Israel.
"Can God's prophets be so unanimous in prophesying against you, if God's Spirit were not joined with them, or if their prophecies were false? The Israelites were 'at ease,' not believing that God was with the prophets in their denunciations of coming ruin to the nation." - Jamieson Fausset and Brown. They go on to say, "The prophets, and all servants of God, can have no fellowship with the ungodly," which is more in line with Poole's interpretation.
Amos 3:3 appears to be talking about the "separation" of the people of Israel from Jehovah because of their sin, and the impending judgment of God for that reason. It is not about "separation" from the liberals, the ecumenical movement and worldliness - I believe in that kind of separation, but it would be inappropriate to use Amos 3:3 as a proof-text for that. Likewise, Amos 3:7 is misused and misquoted today by those who say we can foretell the timing of the Rapture. That is not what Amos is talking about - the "secret" revealed here is the approaching judgment of God upon Israel in 722 BC.
