My young kids have expressed interest in playing Heroes of the Storm, but I'm a little concerned about the swearing in the game. The ESRB description contains swearing that I'd like to limit at their age. Mostly it seems to be banter / dialogue. Also, I've noticed Tycus swears, but I'm not sure when. Can that be turned off?
There is a Mature Language Filter option in the game. You can enable it by opening the options menu, selecting the "Social" tab and then checking the box labeled "enable mature language filter".
Note: I am basing this on the following video:. It might be possible that the options have changed since then.
As Nzall states, there is a mature language filter for chat to censor things your teammates say, however I believe you are specifically referring to the announcer's voice lines. If you got to Options -> Sounds, you can disable/turn off the announcer by clicking the check mark next to it. Most of the things the announcer states also have visual cues as well, so you won't be lacking too much, you will just have to pay more attention to the screen.
There also a set of new announcers that you can unlock from boxes or with shards. I haven't used one yet, however they may have different voice lines, and there may be one who doesn't swear, or swears less. This may be a better option if you can find such an announcer as you will still have the audio cues and won't have the swearing.