Ok, some notes first:

There is no need to use a table for something that is just one single cell. You're also not closing the table in the first switch result, by the way.

There is also no reason for those css classes currently present if they're not defined to do something anywhere.

I would make this two templates, the main template that evaluates the dates, and a subtemplate for the output:

Main template:

Template:Timeline

{{#switch: {{CURRENTMONTH}}-{{CURRENTDAY}}
| 11-29 | 11-30 | 11-31 | 12-01
| 12-03 | 12-04 | 12-05 | 12-06
| 12-02 = {{timeline/box|Yoon Bum|December 2nd|Yoonbum Birthday.jpg}}
| 01-23 | 01-24 | 01-25 | 01-26
| 01-28 | 01-29 | 01-30 | 01-31
| 01-27 = {{timeline/box|Oh Sangwoo|January 27th|Sangwoo Birthday.jpg}}
}}

Subtemplate:

Template:Timeline/box

<div style="border:2px solid #808080; width:100%; text-align:center; border-radius:10px; padding:10px; margin-bottom:20px;">
<h2 style="color:#808080; margin:0 auto;">Happy Birthday, [[{{{1}}}]]!</h2>
''{{{2}}}''
[[File:{{{3}}}|265px|center|link={{{1}}}]]
</div>

• The main template calls the subtemplate, passing character/pagename (1), birthday (2), and the filename(3).

• Timeline/box uses the same layout as your mainpage boxes. Depending on where on the main page you use that, you may have to adjust the css styles a little.

• In case the character name and its pagename is not always the same, you may have to add another parameter for the subtemplate.

Add </noinclude> before the switch:

</noinclude>{{#switch:

Thank you sm!!