There is no way to use contextual links on mobile devices at the moment, because they are only triggered by hovering your mouse over the editable/configurable items, and mobile devices work with taps.
In D8, they have added an "Edit" button at the far right of the admin bar, that toggles contextual links on/off:

We should have the same.
GitHub Issue #:
3417
Recent comments
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Problems with HTML content and text formats
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Problems with HTML content and text formats
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Problems with HTML content and text formats