It depends what you're doing with that stream, most basically you would create a nano/micro EC2 instance that will just trigger Lambda events on every new tweet. Or you could create some more intricate script that does a lot of pre-processing and then stores it in RDS or S3, and with each new update to either of those sources kick off a Lambda.
Unless the API can stream directly into one of those sources you'd probably need a long-running process, perhaps running on a CaaS like AWS Fargate.
I guess you could argue where to draw the "serverless" line, at functions or containers, but Zeit is calling this container service "serverless" so I think Fargate would fall into the same category. I think it would make sense for Zeit to eventually support long-running containers too (looks like the current max is 30 minutes, I'm not sure how they chose that number)