The group of vegetables that is "peppers" encompasses a wide variety of peppers. While most of the peppers we know and love (or fear) today have South American origins, now peppers of some kind can be found in almost every world cuisine, on every part of the menu from appetizers to desserts and drinks.
For the record, peppers and pepper are not the same thing. Table pepper, or peppercorn, whether black, white, pink, green, or sezchuan, comes from a completely different plant. Peppercorn is safe for people who can't eat nightshades.
Because of their light flavor, peppers are easy to replace in almost any recipe. I'll divide this replacement guide into two parts: replacing bell peppers, and replacing hot peppers.
Replacing Bell Peppers
Bell peppers come in a variety of colors, including green, red, orange, yellow, white, and purple. Green bell peppers are actually less-ripe red peppers. They are crisp and watery when raw, and have a light, subtle flavor that, when ripe, is almost sweet.
To replace chopped or sliced bell peppers in raw recipes, consider using a fresh cucumber, seeded if you like. For small chunks (like chopped small in a pasta salad), you can use zucchini or yellow squash. Also in salads, you can use diced, sliced, or julienned carrots or celery. All of these provide a watery crunch without too much extra flavor (though celery and cucumber would add a bit of extra flavor).
For cooking, remember zucchini, your best friend for replacing eggplant?
Other substitutions for bell peppers in cooking include carrots and celery, as I mentioned above, and the stems from leaves of chard. That's right! You can save chard stems and chop them up to use in various dishes. Pink chard stems will leak some of their color in a dish, though, so your stir fry might get a little bit of a blush.
Because their delicate flavor usually cooks down and drowns in a strong sauce, bell peppers don't usually lend a lot of flavor to a dish, so often skipping them altogether is fine.
For people missing fire-roasted red peppers: I've never tried fire-roasting a zucchini, but I rather suspect it'd be almost as awesome as fire-roasted peppers.
Finally, many people (including restaurant servers, ugh) forget that Paprika is powdered red pepper. Often, it is only in recipes for garnish, color, or for an extra "oomph" that you may not need, so if it is a small amount (1/4 tsp or less), leave it out. Ground pink peppercorn is a pretty replacement if it's just for looks. You can also consider using a tiny bit of tamari to up the umami factor of your dish. If there is more than 1/4 tsp paprika, consider what the paprika is doing in the dish. Is it adding a tiny bit of sweetness and/or warmth? If so, add a tiny amount (I'm serious. Tiny. Smaller than a pinch) of ground cloves. Cloves lend a hint of warmth and sweetness to a dish--but a little goes a long way! If a recipe calls for smoked paprika, it's usually just for a smoky flavor. Invest in a bottle of liquid smoke (it's cheap and totally worth it), and use that instead of smoked paprika. Replacing it one-for-one usually does the trick (ie, 1 tsp liquid smoke = 1 tsp smoked paprika), but taste test and add more liquid smoke if you like things smoky. (An additional quick warning: paprika, sometimes listed as "paprika oleoresin," is often used as a coloring agent in red or pink things, particularly meat substitutes, so watch out for it!)
Replacing Hot/Chili Peppers
Lots of people who can eat nightshades still can't or don't eat hot peppers. Because chilis are used so sparingly in cooking anyway, if you don't like spicy things, you don't need to substitute for hot peppers at all: just leave them out. Volume-wise, they aren't a significant percentage of the final dish.
However, a lot of us like(d) spicy foods and miss hot sauce terribly. For people like me, who used to pour hot sauce on everything and add extra cayenne to all her curries, replacing hot peppers is a necessity.
First of all, buy some high-quality pepper. I'm not one of those snobs who will only tolerate freshly-ground pepper, but if you are going to use store-bought ground pepper, you want it fresh, and you want it strong. It's still relatively inexpensive, and it makes a difference. Add at least a little black pepper to any dish you want spicy; you can add generous amounts to Chinese, Caribbean, and Southern dishes.
But black pepper isn't the only way to go. There are lots of other varieties of peppercorn, and each has a distinct flavor. Check out Frontier's "Gourmet Peppers of the World" on their website. I think most major spice brands carry varieties of pepper like this. They're a little more pricy than normal black peppercorns, but cheaper than bottles of hot sauce would be. I've tried a lot of different flavors and they're all really interesting, but if you want heat that can replace cayenne or other chilis, you'll want Cubeb, Sezchuan, Longberry (okay, I haven't actually tried this, but the Frontier website says it's the spiciest!) and maybe even Pink, which isn't as hot but has a warm taste that, like cloves, gives the impression of heat.
Okay, so, ADD PEPPER. But pepper has a distinct flavor of its own, so I know, that can't be your only option. Here's where we come back to cloves. Like I said when I was talking about paprika, cloves in a dish intimate sweetness and heat, though they're not actually sweet. They are rather hot for a spice. Use ground cloves sparingly, but use them in South American (I always put it in my guacamole, in taco fillings, and in nacho-cheese-style sauces), Indian, and Middle Eastern dishes for extra warmth in a dish. Use a tiiiiiny bit at first (again, less than a pinch), then ramp up as long as the flavor is not overpowering anything else.
Other ingredients that add warmth to a dish are ginger, garlic, mustard powder, horseradish, and wasabi. Since most dishes that call for hot peppers also use garlic, try adding more garlic. For dishes that call for ginger already, try adding a little more of it near the end of the cooking time; the fresher it is, the more of a bite it will have. Dried mustard, horseradish, and wasabi can also burn you in that nice sinus-clearning way, but they have really distinctive flavors, so use them with care. (You can up the amount in a dish if they're already there, otherwise I might not try it.)
Certain people (hi, mom!) hate when recipes call for things "to taste," but since these aim to replace heat, and everyone has a different preference of how "hot" something tastes, replace "to taste." If a recipe calls for cayenne, try a 1-to-1 replacement with pepper and a tiny pinch of clove; taste test and work up from there. If a recipe calls for hot sauce, use these replacements, but also add a dash of vinegar or lemon juice, since hot sauces are usually pretty acidic.
Finally, dried peppers like chipotles, pasillas, anchos, and serranos are often dried in smoke houses, or using smoke. If you are dealing with a recipe that calls for one of these, add a tiny bit of liquid smoke (I'm telling you, it is awesome), like 1/4 to 1/2 tsp, to get a more complex flavor. Do this even if you're not bothering with the other replacements; it'll still work wonders for your dish.
If you can't have nightshades, you can't have hot sauce, and you're not going to get that same eye-watering, face-flushing shock that chilis can give you. But you don't have to go heat-less!
One final warning: processed foods often list "spices" in their ingredients, and often, "spices" includes paprika, cayenne or chili pepper flakes. Always read ingredient labels, and try to stay away from anything that lists the mysterious "spices."