The thing about racism is that you really can't portray it accurately without making it frustrating for the players involved. In most cases, there's nothing wrong with scaling it down to the point that you can give the players an idea of what racism is like and still make the campaign understandable and playable ... think about the standard Hollywood script as an example of this. (People from different races unite against a common enemy and win, learning in the process that everyone is equal.)
I think the key in those cases is to not hinder the story. Unlike in real-life situations, you'll always need to provide a path for the story to continue. If shopkeepers won't deal with Warforged, then there needs to be an NPC somewhere in the city who will. If the party can't find a place to stay, then you need to have a sympathizer somewhere who will share their floor. If the party can't find "work", someone needs to overhear their conversation and explain the situation while suggesting a possible solution. (The lord's son was killed outside a bar last month; witnesses pointed to a Warforged, even though he wasn't the guilty party. Finding the perpetrator and getting him to confess might open up some quests even if it doesn't completely change the lord's mind.)
Remember that with rare exceptions, reactions with a town will be mixed. Some townspeople will be sympathizers, even if not openly. Not all Warforged will be interested in fighting to gain true equality, even if only because they simply can't believe it will happen. Having said that, don't make the party search for the one person in 100 who'll help them; make it clear from the first conversation what the general attitude is. ("You get the idea that most people in the marketplace are bothered by your presence; some appear to be neutral, but you can't tell if they'd take your side or if they just want to stay out of it. As you head past the last stall, someone in the alleyway just beyond clears her throat. 'So your money's no good here?' she asks.")
Even scaled down to playability, racism should never disappear. If you do find the true killer, the lord will not suddenly accept a Warforged in his court. If your party fights off a wave of bandits that has been picking off merchant caravans, there will still be merchants who won't trade with you or who will charge higher prices. Make sure the party is always aware of racism in this setting ... but again, make sure they always have a way to accomplish what they need to do.
In some cases, though, you simply won't be able to introduce racism successfully, particularly if you're not playing in a real-life homogenous group. It's easy to inadvertently offend someone from a different race by trying to portray racism within the campaign: after all, it's entirely possible that they've experienced much worse situations in real life than you'd want to depict in-game, and they might not be interested in roleplaying situations that they experience in real life (even if they're not actually playing Warforged characters).
I've only played in groups where everyone's real-life backgrounds were similar, so I can't say from experience how to handle the latter situation. The best suggestion I can make is to be especially sensitive to your players' concerns with respect to Eberron. If one or more players seem reluctant to play in that setting, talk with them after the group has left (and ideally before you actually start play) to discuss their concerns.
Remember, the most important thing is that everybody has fun. Maybe that means that Eberron isn't the right setting for your group: if that's the case, put it aside and find another setting.