Unless you have specifically configured TVHeadend to 'drop' the subtitles they are enabled by default - TVHeadend simply passes through (or re-encodes) the whole transport stream so it's up to the client to decode the subtitle data.
Which stream profile are you using? If it's HTSP then subtitles are passed through by default (no way to change this).
If it's anything else, from the TVHeadend webui, switch to the Configuration > Stream > Stream Profiles tab, pick the relevant stream profile and ensure that 'Subtitle codec' is set to 'Copy codec type'.
If that's all correct then it may simply be;
The program you are watching does not have subtitles (see below)
The format of the subtitles in the program you are watching isn't supported by the client
To check if there are subtitles in your current stream:
Go to Configuration > DVB Inputs > Services, filter the 'Channel' column by the channel that you are trying to watch, click the 'i' icon and ensure that the BOTTOM section of the screen has at least one TELETEXT or TEXTSUB entry.
Sometimes though, the combination of container, codec, subtitle and player just doesn't work. For example, TVHClient on my iPhone doesn't support DVB subtitles when used in a matroska container format.