Sorry in advance for the ramble.
For exclusion, the New Testament book that came closest to being left out of the canon is undoubtedly the Apocalypse of John, aka the Book of Revelation. Getting it excluded from the canon certainly would have interesting effects on Christian eschatology, but not much else-- Revelation was never especially popular or well-read before the 1800s. Though of course this could still cause all sorts of crazy butterflies. Fewer Christian doomsday cults would be nice, I guess.
For other exclusions, let's use the Muratorian fragment's canon c. 170, Codex Vaticanus's canon c. 310 and Eusebius's list of disputed letters c. 325.
Here's a list of disputed NT books, in order of likelihood of exclusion. No spacing between two lines means about equally likely.
-James (listed by Eusebius as disputed c. 325, excluded by the Muratorian fragment's canon c. 170)
-2 Peter (Same as James, Eusebius disputed list and excluded by Muratorian)
-Jude (Eusebius lists, Muratorian includes)
-2+3 John (Eusebius disputed list, maybe in Muratorian)
-Hebrews (Muratorian excludes)
-1-2 Timothy (Vaticanus excludes, generally accepted otherwise)
-Titus (Vaticanus excludes, generally accepted otherwise)
-Philemon (Vaticanus excludes, generally accepted otherwise)
-1 Peter (Muratorian excludes, generally accepted otherwise)
-1 John (Probably but not definitely included by Muratorian, generally accepted otherwise)
Also, the heretic Marcion wrote the earliest known canon c. 130, only including Romans, Galatians, 1-2 Corinthians, Philippians, Colossians, possibly Ephesians, 1-2 Thess, Philemon, and a different revised(?) version of Luke. Take of that what you will.
Now for the consequences.
Excluding James could have big effects on soteriology-- it was an essential prooftext for Catholic apologists arguing against sola fide during the Reformation.
Jude and 2 Peter don't matter much, besides being the only place in the NT that explicitly set out that Satan and his angels are imprisoned in Tartarus.
2 John and 3 John similarly don't really matter-- they're very short and don't say much unique.
Hebrews, opposite to James, was part of the core of the Protestant argument in favor of sola fide. Otherwise unimportant.
1-2 Timothy have the only explanation of divine inspiration in the Bible, plus THE key verse excluding women from preaching or holding authority over men. Big implications.
Titus is not very important. Very short, mostly just about church governance.
Philemon has potential implications for debates over slavery, but it's unclear which side it's exclusion would help.
1 Peter contains the only explicit reference to the Harrowing of Hell, where Jesus preached to the dead after his death. It also says women should submit to their husbands.
1 John isn't very important, except for a later pro-Trinity addition. Might help non-Trinitarianism, but that sort of addition could happen to another book of the NT just as easily.
For adding a book to the NT, I'm less sure. But here's a list of what you could probably get in, listed in no particular order. Not sure about which of these would be the easiest to get in.
-Shepherd of Hermas: A story about a former slave who converts to Christianity, plus lots of weird stuff. Possibly adoptionist, possibly binatarian, though not definitively. Also pro-Law of Moses, but so are Matthew and James and they got in, so whatever. Synergist (works and faith) soteriology, says you only have one chance for pennance if you fall into sin post-baptism(?????). Could have some strange consequences if added.
-Apocalypse of Peter: Early Dante's Inferno where Peter is given a guided tour of heaven and hell. Mostly focuses on hell, and graphically describes tortures there. Add if you want Christians to be more sadistic and hell-focused, I guess?
-Epistle of Barnabas: Anti-Jewish polemic, allegedly written by the apostle Barnabas. Heavily allegorizes and reinterprets laws of the Old Testament, could lead to Christians taking the Old Testament less seriously. Also could make Christianity more anti-Jewish, obviously.
-The Didache: Basically a short statement about what Christianity is and some rituals. Jesus called the servant of the father, kinda Arian christology (technically subordinationist but whatever). Also potentially supports a less centralized Church structure.
-Acts of Paul: All about the acts of Paul, self-explantatory. Encourages women to preach and baptize, would be a Big Deal for Christian views on women. Also says Paul was beheaded but milk instead of blood came out, which is pretty funny.
-Gospel of the Hebrews: A Jewish Christian gospel that unfortunately didn't survive in full to the present day. Apparently thought the Holy Spirit was the mother of Jesus(?). Also kinda Gnostic despite being Jewish-Christian?
-1+2 Clement: Written by the apostle Clement. Long and rambly but not clear exactly what it would add doctrine-wise. I don't know too much about these so would appreciate if anyone knows more.