1 and 2) "в" is not applicable to web-site or web-page in these contextes. You must use "на".
I think it is because we observe it on display, and "in/inside display" is nonsense.
However, if you use "на" it is ok. Also we can really change grammatical case to:
Нам следует разместить новый материал на нашу веб-страницу.

3) Он ответственен за введение данных клиентов в нашей компьютерной программе.
Is ok.
Also can be:
Он ответственен за введение данных клиентов в нашу компьютерную программу.

As was said above there is little shift in meanings. A very thin one. It's hard for me to express it...
First one answers question "where inputing of data takes place? in our program (as workplace)." (где?)
Second one answers question "where data is placed to? into our program (as database)." (куда?)

Russian "где" and "куда" translates to english as "where", but in russian they stronlgy differs.