You are not going to find anything outside the chili family that gives quite the same flavor, so substituting flavor-wise is not going to be possible.
Note that paprika is a spice ground from particular pepper, so if you are allergic to all capsicum peppers, you don't want to use it.
What you can do is build other flavorful combinations which you enjoy and which you can eat.
Some things to consider that bring a touch of some type of heat with them include: Mustard powder (a touch of mustard-type heat, and a deep flavor.
Probably want to combine with some herbs like oregano or rosemary.
Horseradish (just a touch for piquancy)
Ginger, which will work very well in Asian inspired dishes, and in combination with those flavors Szechaun peppercorns (not a capsicum pepper)
-- they have a unique flavor and effect; you will have to decide if you like them
All of these flavors are "hot" but from different chemicals than the capsicum peppers, so they will all have different effects.
cooking.stackexchange.com/users/14401/SAJ14SAJ
I share your allergy and have for some time.
First - I'm very sorry, it's not a fun one to have.
Second - there are a lot of spices you can use that give color and flavor without going into the pepper family.
I have a recipe for a curry powder you can use:2
tablespoons whole cumin seeds, toasted2 tablespoons whole cardamom seeds, toasted2 tablespoons
whole coriander seeds, toasted1/4 cup ground turmeric1 tablespoon dry mustard
It's adapted from an Alton Brown recipe (no cayenne)
but the cumin still adds a bit of a kick without requiring an epi-pen.
Also - you can use wasabi in some cooking
- it gives a bit of a bite as well.
I also use tumeric, as it adds color as well as flavor.
I use quite a bit of raw garlic and onion, as it gives a bit of a bite to food, but too much will leave a casserole bitter.
Hope this helps!
cooking.stackexchange.com/users/24339/user24339
Galangal root is a possibility ( more info ).
It's sort of like ginger that's been kicked up a notch on the hot/spicy axis.
Chinese, Vietnamese and Thai stores will have it.
There's also a powdered form available online.
I've never tried that, but maybe it doesn't suffer the same terrible fate as powdered ginger.
Prickly
ash (Sichuan pepper) and Japanese Prickly ash are other possibilities.
The Sichuan pepper does add a funny sort of heat to things.
I've never found the Japanese version to try.
cooking.stackexchange.com/users/5455/Wayfaring Stranger
Thanks for all the tips.
I have arthritis and any thing in the 'deadly night shade family of plants is bad for us.
That is: Potatoes, tomatoes, peppers, aubergines, chilli and its sidekick Paprika.
These two latter things make my joints swell - not life threatening but very painful.
In looking for an alternative, I have found some Garam Masala does not contain any Chilli or Paprika and many recipes for Korma curry do not use chilli.
It doesn't quite have the kick but it is very tasty.
cooking.stackexchange.com/users/65749/Ashfish47
Be very careful with mustard.
Premade mustard contains paprika.
I make my own for recipes that call for it, and use the dry mustard to cook often.
Cumin is a wonderful way to add flavor, and I especially like corriander.
Experiment and see what you like best.
Its a bit of a crapshoot and everyone's tastes are a little different.
Cook in small batches, so there's no leftovers, until you find what you like best.
Good luck!
cooking.stackexchange.com/users/68310/Ciaodhna 
Black Pepper falls into a very different plant family than chilies and bell peppers.
Also the compound that produces the spicyness is different (piperine vs capsaicin).
There's are a A LOT of different pepper species that should offer some variety if you can get a hold of them.
Also don't underestimate the differences in black pepper alone.
This site also lists a few other species that are sold as "pepper" but are neither Solanaceae nor Piperaceae: Schinus Molle
/ Pink pepper Prickly Ash
/ Sichuan pepper
And as others have suggested, the Ginger family is always a good source of flavorfull spicyness.
Further options for spicyness taken from
this answer : Horseradish Mustard Or Wasabi as others have also already suggested and warned about fakes.
cooking.stackexchange.com/users/61049/Kempeth
I'm allergic to red chili pepper and usually make do with ginger, mustard and some cumin.
cooking.stackexchange.com/users/67551/Slothful Slob
I recently learned that I am allergic to ALL forms of pepper except black pepper.  Bell peppers, chili peppers, etc.  I am allergic to and it makes my throat swell up.  I have several recipes using ground chili pepper.  Is there a substitute spice that will give my (mostly ground beef casseroles) recipes SOME flavor?
Chili and bell pepper substitute due to allergies?
cooking.stackexchange.com
<substitutions><spices><allergy><chili-peppers><bell-peppers>
cooking.stackexchange.com/questions/33534
cooking.stackexchange.com/users/17877/Allergy Plagued
Ginger, mustard, horseradish, Szechuan peppers, wasabi and galangal roots are possibilities that add heat to some dishes. You can try mustard powder, horseradish, ginger, wasabi, galangal root and cumin.
You can try mustard powder, horseradish, ginger, wasabi, galangal root and cumin.