On the Battle.net forums, Overwatch Game Director Jeff Kaplan revealed that both a replay system and a spectate only feature are still on their schedule, but have been bumped due to work on the high bandwidth server technology, Season 2 competitive mode changes, and more:

Both have been on our schedule for some time but have gotten bumped for different reasons. The same people who would work on those features ended up being re-prioritized to working on high bandwidth server technology, Season 2 competitive mode changes and matchmaking improvements. We also have another feature request in front of the replay system — the ability to save highlights. Some of the work here is a step towards replays (another top requested feature) so it makes sense for this to come first when the time comes.
Kaplan didn’t give an ETA on the new features, only saying, “We’re as excited as you guys are for these features. We’re really hoping to get time on them.”
As for those high bandwidth servers, Community Manager Lylirra explained that they’re “currently in the process of rolling out high bandwidth support globally for PC.” It may take a few weeks, “but once fully deployed all games including those in Quick Play and Competitive Play will be running in high bandwidth mode by default.”
The high bandwidth option adjust the game’s client update rate from 21 updates-per-second to 63 updates-per-second, making the game feel m...