Consider that the bride has (usually) just spent hours having her makeup carefully applied. Yes, it is romantic when tears come to her eyes; that means she is fully engaged in the process of the wedding, and something has moved her deeply. At the same time, she does not want her mascara running down her cheeks, nor does she want her nose to run. Help her retain her maximum photographic advantage. Whip out your hanky and let her delicately blot the tears away.
During my dozen years of officiating, I have also used that hanky to swish away enthusiastic bees from a bride's fresh-flower headdress, and to help a bride mop away the "glow" from marrying on a hundred-plus degree summer day.
The groom is just as likely to tear up. He often has a nicely folded handkerchief in his breast pocket, but yours is more convenient while he reads his vow.
Make sure you are not emotionally invested in the handkerchief, because in the ensuing chaos after the ceremony, it won't make its way back to you.
I remember performing a wedding in a white-towered tent surrounded by a blooming lavender field. The bride had thoughtfully placed a table piled with vintage floral hankies at the entrance to the tent. Alas, there was no sign specifying that guests were able to choose and have one, and so most of those hankies went unused that day. It must have taken a lot of work to assemble that lovely heap of cotton wisps; I hope the couple later made something like a hankie quilt with them.