One should update template on Sharepoint server to make its changes visible/available to Sharepoint, i.e through Infopath Form Web Part.
If you want to change choices without changing the template then you should define choices outside of the template itself. So change (in Drop Down List Box Properties) the List Box Choices from "Enter choices manually" to "Get Choices from an external data source" having configured the "Data Source:" (Add... button). One of the easy variants is to create and use a sharepoint list column (in another or the same sharepoint list) for this.
Fig.1. Get choices from an external data source (in Drop-Down List Box Properties). I've used here as datasource the Title field of another (could be the same) sharepoint list called Dropdown
Answer to comment:
You did not provide details how you created the Infopath form but most probably, if you created the form by pressing Customize Form button, the choices were embedded in the form template itself
Fig.2. Enter choices manually (in Drop-Down List Box Properties).
Which are filled indirectly through embedded into Infopath template file choices.xml, Fig.2a
Fig.2a. Uneditable locked Choices data connection in Infopath template auto-generated from sharepoint list column of type choice
That this file choices.xml is embedded in template one can make sure by exporting source files (in Infpath Designer 2010, File > Publish > Export Source Files)
And to enable choices to change independently on Infopath template in order to avoid template republishing for updating (without Infopath template changes through updating embedded choices.xml), you can get/link choices from(to) an external data source. There are variants, but the most quick and simple one is to create and use another sharepoint list for this (cf. Fig.1)
After changing data source of drop-down list box, make sure to delete old manual defaults in Infoapth template. In Infopath Designer 2010, Data tab > Default Values button with green checkmark (in Sharepoint Form Data section on ribbon)
Fig.3. Delete remaining embedded into template defaults
So, having created and filled Title colum in Dropdown sharepoint list
Fig.4. Field Title of Dropdown Sharepoint list
linked as data source of choices in drop-down list box Choicesss (with underlying data field with the same name) in Sharepoint List Choices the choices are:
Fig.5. Choicesss drop-down list box in Infopath Sharepoint List Form for sp list Choices
or what is the same, here is a web part aspx page with that inserted Infopath web part
Fig.6. WebPart ASPX WebPage with Inserted Infopath Form Web Part
Fig.7. Infopath Form Web Part Properties
Fig.8. IP Web Part ASPX WebPage after Dropdown sharepoint list update (without Infopath form template republishing)
Fig.9. Updated Sharepoint list Dropdown
Reply to comment:
I think it is the default data connection 'Choices Data Connection'
Fig.10. "Link photo 1" from comment
Fig 11. "Link photo 2" from comment
By default it get's hes choices from choices.xml – mchello
I do not know how you get to situation on Fig.12 since I observe it as in my Fig.2 and what is the sense of it.
But the point is that in both mine and your situation there is embedded into Infopath template file choices.xml and in order to avoid the necessity to update choices of drop-down list box by template republishing, I proposed a workaround of of having data source directly from a sharepoint list column of type "Single line of text"
Have I answered your question how to add/update drop-down list box choices in Infopath template (which is the same as in a webpage with Infopath Form Web Part from it) without necessity to republish the template?