Here are all the ones listed by default, image courtesy of FlappersFlappers and Bulbapedia themselves and information directly from the Pokedex:
Please note that this contains several orangish-reds and more obvious oranges as well, Game Freak did not create a category for orange mons.
Some other ones I thought of that were classified as other colors but could be argued as red:
- Pyroar (mane, classified brown)
- Taillow and Swellow (breast, classified blue)
- Pikipek (head, classified black)
- Zoroark ("hair", classified gray?? Woulda thought black). Possibly Zorua too, but a much lesser degree.
- Florges' red form (all forms are considered white, I think Game Freak was lazy here)
- Larvesta and Volcarona (horns and wings, classified white from their bodies; this "red" is really orange, but if orange goes in red's category, I'll toss this one in)
- Cherubi (whole body, classified pink but it's such a dark pink that it teeters on red for me)
- EDITS: original form of Vulpix (wait, this is classified as h*ckin brown??? e g g s b l a I n, GF!)
- Volcanion (whole body, classified brown)
- Nosepass and Probo (big ol' red-orange noses, blue bodies, but are somehow still classified gray)
- I'm sure some Vivillon is out there that's red.
- Zangoose is half-red and half-white, looks like the Dex sided with white.
- Autumn form Sawsbuck have large red horns, though I don't think it's still possible to get these. (classified brown)
- Arceus, most prominently in Fighting form, but also Fire because we're including orange
- Bounsweet looks just like Cherubi and yet it's classified purple.
- Phanpy has some, too, I guess...but it's really little (classified blue, of course). Donphan by itself is not (classified obviously gray).
- Pirouette Meloetta (head, Aria and Pirouette classified white), though who uses that?
- The first ones that comes to mind are Grubbin and Charjabug, because I raised one.
- Combee and Vespiquen are VERY red - best friend has one.
- Gyarados, obviously.
- Mudbray and Mudsdale are a bright orange - again with the orange thing, and their legs are red anyway so I guess
- Purrloin and Liepard
- Timburr family is much more noticeably red, even if not their primary color, it plays a major part in its design IMHO
- Terrakion has red horns
- Wimpod's purple goes red, much less noticeable when it evolves into Golisopod though
- Glalie's eyes are so bloodshot that it's wrong to ignore it being a primary part of it
- Moltres goes from orange to actually red.
- Cyndaquil line used to get really red along its back, but starting in Gen VI, they toned it down to a much duller brown which makes me sad TBH.
- Ferrothorn. Just Ferrothorn.
- Honedge and Doublade both have blood-red blades. It's much less prevalent on Aegislash, but the edge is still red...
- Skorupi and Drapion.
- Shellder. Just Shellder. Very orange.
- Morelull. Just Morelull. Shame, because this looked good while Shiinotic's looks ugly AF.
- Armaldo. Just Armaldo.
- Linoone's brown streaks are now orange.
- Donphan's trunk and spine are a reddish-orange.
- You would have thought that Elekid and Magby stayed somewhat consistent for their evolution line's shiny colors, until their second forms step in and say "I don't know about that one chief" and goes red and pink, respectively.
- Chespin line's shiny forms are strikingly red.
- Cottonee. Yup, just Cottonee. Reddish-orange.
- If we're including oranges, Swoobat is definitely included. Woobat is by no means red.
- Shroomish and Breloom.
- Before Gen VI rolled around, Palkia was pretty red while now it's more subtle.
- Phantump and Trevenant have large red patches.
- Lugia makes a switch from blue to red.
*I am not including any shinies that hardly change their color that were red in the first place, or appear in the first list. (Nosepass family, Paras family, Kricketot family, Torchic family, etc.)
Alright so that's everything that I could think of. Now it's your turn, PokemonDB, to yell at me for missing your favorite red mon. I will gladly add it in.