In doing an objective view of this situation, it is a close battle betwwen 2 of the rivalries, us against Buffalo, and Washington against Dallas. The others in the list, Piitsburgh/Cleveland a weak 6th(Rams/49ers a lot fiercer rivalry) a rivalry only good during the orage pants days for Cleveland(1975-1983). Packers/Bears 5th A lot of respect between the 80 year rivals, possibly too much. Also, often only one of those teams has been a good team.Us against the Hapless Ones 4th. A rivalry that has become stronger since 1997. The fans of the Hapless one generally do not consider us to be a main rival though, they look at their stadium mates, the Giants as their primary rival. Raiders/Chiefs 3rd. A nasty history between the 2 teams(particulary the 1970 game in Kansas City). However the Raiders have so many other teams that consider them the primary rival that it is not the most intense rivalry.
Now comparing the remaining 2 rivalries. The Washington Dallas rivalry is a extremely intense one. Each team comes from an extremely egotistical city. The team nicknames represent the primary American historical fight of the cowboy against the indian. The color schemes are radically different of the silver against the burgundy and gold. Dallas prior reoputation of the holier-than-thou efficient computerized machine against Washington's gritty old men. Later on with Jerry Jones's obnoxiouness spilling down to his team and Washington becoming the poster team of NFL stability. And many of the games have been classic.
Then on us and Buffalo. The ultimate culture clash. Inefficient industrial town against chic tourist town. Nasty snow against hot sun. Astro turf against lovely grass. Dull ill-fiting uniforms against bright well tailored ones. Fans that really dislike each other. And each teams really enjoys rubbing it in to each other.
It is a tough one to call. However putting aside my emotional pick, I go with the Washington Dallas rivalry as the most intense by a slim margin.