Is it ever competitively viable to use a Pokemon in a lower stage of evolution?

Because most Pokemon increase their stats when evolving, you would not normally see a not-fully-evolved competitively. The most common exception to this is caused by pokemon holding Eviolite:

Effect: Raises the holder's Defense and Special Defense by 50% if the holder is not fully evolved.

It is important to remember that this will multiply not only base stats, but the final value, which includes IVs (Individual Values) and EVs (Effort Values). This is obviously most effective for acting as a wall. There are multiple pokemon which see competitive usage with Eviolite, but most notable are Chansey, and Porygon2. Dusclops, Doublade, Scyther, Golbat, Magneton, and Gligar may also use it well.

For numbers lets look at smogon's OU builds for Chansey and Blissey. Plugging the level (used 50, per VGC), EVs, nature and max IVs into a calculator (links on names below) we get:

HP Atk Def SpAtk SpDef Spd

Blissey: 331 27 68 95 187 75

Chansey: 326 22 62 55 157 70 (without Eviolite)

Chansey: 326 22 93 55 235.5 70 (with Eviolite)

This shows Eviolite Chansey has 137% Defense, and 126% Special Defense compared with Blissey, with 98% HP, 93% Speed, 82% Attack, and 58% Special Attack. This obviously increases survivability, and competitive Chansey would not have a move depending on Attack or Special Attack, so those stats would not matter.

A major downside of using Eviolite is that it is taking up an item slot, meaning no Leftovers or 'choice' items, but many competitions (including VGC) enforce an item clause (no duplicate items), which leaves leftovers for another member of the team.

Threads on smogon and gamefaq cover the eviolite users.

There is generally no competetive reason to use Not Fully Evolved pokemon outside of the Not Fully Evolved and Little Cup tiers (where fully evolved and/or high level pokemon are expressly banned). NFE is it's own category and the fully evolved pokemon only are in the other tiers largely because stats almost exclusively get higher (there are exceptions but they're limited to a select few evolutionary families).

In addition, NFE pokemon also have theoretically more limited move sets; a stage one pokemon can only learn moves from their first evolution (plus egg moves). A stage 3 evolution has 3 stages of evolutions' moves that it could have learned, plus egg moves. Again this depends on the family, but in some cases a full evolution can give access to particularly useful moves. For example it lets every pokemon that can learn TMs have access to Hyper Beam (full evolution is a requirement for the TM), though ironically Hyper Beam isn't actually a great move competetively.

The stat boosts are often quite significant and are the main reason not to use lower-level pokemon, but the reduced moveset is also limiting. As an example of the stat difference, look at the average stats of Dragon types for overall and fully evolved Pokemon. Note that the "overall" stat is actually skewed higher than a "not evolved" distribution would be, and powerful pokemon with a single evolution (including several legendaries) skew the distribution as well, yet fully evolved pokemon still have a clear advantage.

Bulbapedia has some evolution related tier exceptions listed as well; in these cases a lower form of evolution is advantageous, albiet usually in specific situations (for example a Pikachu holding a Light Ball can have higher special attack than a Raichu of the same level).

Having played a lot of competitive battles since writing this question, I have since learned of many Not-Fully-Evolved Pokemon which can be competitively viable due to differing abilities or good (enough) stats. Hell, a Follow-Me Magmar was on a team that won the VGC Worlds Championship in 2013!

Most of them will run Eviolite, which boost a Pokemon's Defense and Special Defense and allow it to compete against it's evolved counterparts. Note that this boost is calculated on the final stats, which includes Natures, IVs and EVs, not just the Pokemon's Base stats!

There are many not-fully evolved Pokemon that are competitively viable in their current form. Here are a couple of the more popular ones as an example:

Chansey

Chansey evoles into Blissey in Generation 2 and up with High Friendship.

Chansey's HP is only 5 base stats less than Blissey (250 vs 255), and with Eviolite doubling it's calculated Defense (which includes Nature and EVs as well as the Base Stats), Chansey is better equipped to deal with Physical Attacks, an area that it's counterpart Blissey is usually lacking, which chooses instead to focus on it's higher Special Defense.

Porygon2

Porygon2 was introduced in Generation 2 as an evolution of Porygon. As of Generation 4 it received it's on evolution in Porygon-Z (when traded with a Dubious Disc).

With Eviolite and Recover, Porygon2 makes a very good wall that can support the team, usually setting up Trick Room and threatening with Ice Beams and Thunderbolts/Discharges. Porygon-Z on the other hand is usually built to be a very offensive Pokemon in a Revenge-killer position

Togetic

Togetic evolves into Togekiss in Gen 4 and above by way of Shiny Stone.

Like Chansey, Togetic focuses on more defensive bulk than it's evolved counterpart, with a lot of Togetics absorbing attacks for their partners with Follow Me. They can also harass with Encore/Thunderwave and support the team with Tailwind.

Murkrow

As of Generation 4, Murkrow evolves into Honchkrow when a Dusk Stone is used on it. The main reason why Murkrow can be competitively viable is in it's Hidden Ability: Prankster, which gives Status moves used by it a +1 Priority. With it's access to a wide variety of Status moves, Murkrow to act as a viable Support Pokemon, especially in doubles.

It can:

• Speed-control with Thunder Wave/Tailwind/Quash
• Harass with Taunt/Torment/Swagger/Flatter/Confuse Ray/Feather Dance
• Set up (or cancel out) weather effects with Rain Dance/Sunny Day
• Stall out opponents with Toxic/Roost Paired with Substitute or Protect
• Perish-Trap with Mean Look/Perish Song
• In a pinch, it can also hit back with Foul Play or Sucker Punch

These are just some of the Not-Fully Evolved Pokemon that can hold their own in a competitive situation, there is also Rhydon, Magmar, Electabuzz, Clefairy, Golbat, Scyther, Doublade, Piloswine and many more.