On April 27, 2013, Justice Cunningham was dubbed "Mr. Irrelevant", the title given to the last player selected in the annual NFL draft. As such, expectations for him in the NFL were limited at best. That being said, you don't get drafted if you don't have the talent to hack it in the NFL. Cunningham is known for his blocking skills, which is good news for him as the Rams starting TE, Jared Cook, is infamous for his lack of blocking ability.
As mentioned above, Jared Cook is the Rams starting TE. Behind him, Lance Kendricks is entrenched as Cook's primary backup and also fills in when the Rams line up in 2 TE formations. After Kendricks, you have the most versatile TE on the roster in Cory Harkey. In addition to filling in when needed at TE, he acts as the primary Fullback in 2 RB formations. It would be a surprise if any of these three weren't on the final 53-man roster when week 1 rolls around.
That leaves the remaining 3 TEs competing for a single roster spot (assuming the Rams carry a 4th TE). In addition to Cunningham, well rounded Undrafted Free Agent Alex Bayer and former Oakland Raiders Defensive End (yes, Defensive End) Mason Brodine whom the Rams are attempting to convert to TE.
Nick Wagoner from ESPN believes that the Rams "have high hopes for Cunningham." That leaves a lot of room for interpretation as we aren't sure if those "high hopes" are short-term or long-term. The highest expectation that could reasonably be levied on Cunningham is that he makes the final 53-man roster and contributes on special teams. Expecting anything more would be unrealistic.
Will he Make the Team?
As of now, Cunningham is still very raw as a receiver and will make his money as a blocker in goal-line sets, much like Mike McNeill has over the past 2 years. However, for the first time since the Greatest Show on Turf days, the Rams have considerable offensive line depth. You could certainly argue that Sean Hooey or Mike Person could fill that role just as easily, freeing up a roster spot to be used elsewhere on the team. Considering that he lacks the athleticism of Brodine and Bayer, it would be a surprise to see him make the final 53-man roster. He is a likely practice squad member if he can clear waivers after training camp.