I’m afraid this is not a bug—at least, not an issue Doom or doom-modeline has any power over, so I’ve moved this to #support:upstream.
Emacs shrinks the mode-line to the height of the shortest character displayed in it. Normally, this is the height of your default font, but icons and bitmaps can be any arbitrary height. The doom-modeline package (which I don’t maintain, btw) uses a dynamically sized bitmap to enforce a minimum size, but only beyond the height of your default font.
Turning on flycheck-mode turns on the flycheck segment of the mode-line, where it displays an icon next to the issue count.
Here are two options to work around this:
Turn off icons in the mode-line altogether with:
;;; add to $DOOMDIR/config.el
(setq doom-modeline-icon nil)
Tweak all-the-icons-scale-factor (1.2 by default) to resize all icons used in the mode-line (and elsewhere):
;;; add to $DOOMDIR/config.el
(setq all-the-icons-scale-factor 1.1)
Alternatively, you could modify doom-modeline’s flycheck segment or advise doom-modeline-update-flycheck-icon to use an icon that isn’t so tall.
In any case, you may have better luck reporting this directly to doom-modeline instead.