I have spent the last year and a half building a completely serverless production service on Lambda, API Gateway, and DynamoDB (along with the standard auxiliary services like CW, SNS, Route53, S3, CF, X-Ray, etc.). It was a lot of work establishing new patterns for many of the operational aspects, particularly custom CW metrics and A/B deployments with Lambda traffic shifting, but in the end everything is set up nicely and I'm quite pleased with the end result. We're starting to ramp up traffic now by orders of magnitude (with many more to come) and it's soooooo awesome knowing the stack is pretty much bombproof. Another super-nice thing is all internal authentication and networking being controlled by IAM rather than security groups/VPC/traditional networking - that aspect alone eliminates a tremendous number of headaches.
My biggest complaints are probably DynamoDB eventual consistency (unavoidable when using GSIs), occasional CloudFormation funkiness (though no urgent prod issues yet, thankfully), CodeDeploy CW alarm rollback jankiness (which doesn't tell you which alarm triggered a rollback!!), and lambda coldstarts. But none of these are too terribly concerning and I have faith they'll get incrementally better over time, hopefully.
The biggest cautionary tip I have is we run all our Lambdas with the max 3GB memory both for peace-of-mind and because the underlying EC2 instances have significantly faster CPU. We were seeing weird timeouts and latency initially with Another cost concern I should also mention is that we mitigate cold starts by running multiple canaries using scheduled lambdas (in addition to the standard canary role of generating a baseline of metrics and immediately detecting/alarming on end-to-end issues). We are effectively maintaining a constant warm pool which, in theory anyway, greatly decreases the chances customer traffic will hit cold starts. I'm not intimately involved with the financial aspects but I suspect achieving the same effect with EC2 would be significantly cheaper, at least with respect to infrastructure costs. I would guess, though, that the developer time savings achieved by massively reduced ops burden and overall system simplicity are probably comparable to the increased infrastructure cost, and very possibly hugely outweighing it.