Where does scripture differentiate between a visible and invisible church?

When Christ speaks of the “church” (i.e. Matt 16:18), He is speaking to believers. But in the Revelation study, Pastor Armstrong expounds that the letters to the "churches" were written to "the fellowship", indicating that they can be made up of both believers and non-believers. Is this a misinterpretation?

You are right to be concerned with whether a teacher’s interpretation aligns with Scripture and is consistent in approach. In his Revelation teaching, Pastor Armstrong relies on the context of the letters to guide his interpretation, and he finds it’s evident that sometimes Jesus is speaking to unbelievers who think themselves to be in the church:

In particular, the churches of Sardis and especially Laodicea receive warnings that suggest an audience of unbelievers within the gathering. Notice the following statements to each:

Rev. 3:4 ‘But you have a few people in Sardis who have not soiled their garments; and they will walk with Me in white, for they are worthy.
Rev. 3:5 ‘He who overcomes will thus be clothed in white garments; and I will not erase his name from the book of life, and I will confess his name before My Father and before His angels.

Rev. 3:15 ‘I know your deeds, that you are neither cold nor hot; I wish that you were cold or hot.
Rev. 3:16 ‘So because you are lukewarm, and neither hot nor cold, I will spit you out of My mouth.
Rev. 3:17 ‘Because you say, “I am rich, and have become wealthy, and have need of nothing,” and you do not know that you are wretched and miserable and poor and blind and naked,
Rev. 3:18 I advise you to buy from Me gold refined by fire so that you may become rich, and white garments so that you may clothe yourself, and that the shame of your nakedness will not be revealed; and eye salve to anoint your eyes so that you may see.
Rev. 3:19 ‘Those whom I love, I reprove and discipline; therefore be zealous and repent.

To the church in Sardis, Jesus says there are but a “few” who “walk with Me in white,” which are symbols of salvation. These “few” are the overcomers, Jesus says, and they are in the book of life. This statement suggests strongly that the rest are not saved, though they are in the church of Sardis also.

And to the church in Laodicea, Jesus says the church is self-deceived thinking it has what it does not possess (i.e., salvation by grace). Jesus uses common symbols of salvation like seeing and white garments again to represent salvation, leaving us to conclude that the church was largely unbelieving.

Therefore, Pastor’s Armstrong interpretation is consistent with the text, not against the text. By the context, we see the word “church” being used in a broader sense here than elsewhere in the Bible.