This is a very interesting and highly important question. Every Jew has a
mesorah (tradition) of how he is to act. This mesorah includes many aspects
of Judaism including, but not limited to, dress, customs, nussach (text of
prayers), diction and pronunciation. Each Jew has to follow his mesorah. I
heard a tape once by Rabbi Yissochor Frand where he states that every Jew
has a "key" into the world-to-come. This key is his mesorah. Therefore, it
is clear that one may not pick and choose his form of pronunciation.
The movement which started in the 50's and 60's whereby Ashkenazic (mainly
in America) Jews started to speak "Israeli" is very difficult to support. I
have heard that an Ashkenazic Jew who pronounces the name of G-d (Aleph,
Daled, Nun and Yud) in the "Israeli" way has not fulfilled his obligation.
This comes into play when one is making brochos (blessings) or when one is
davening (praying). This also has a huge impact when the speaker is a Baal
Koray (Public Reader from the Torah) or, to a lesser extent, a Chazan
(Synagogue Service Leader). In these cases, not only has the speaker not
fulfilled his obligation but neither has the listener. I have been in one
Shul where the Ashkenazic Baal Koray (who was a born Israeli, by the way)
pronounces every word in the Israeli pronunciation except for the name of
G-d which he pronounced Ashkenazic. I am pretty sure that he did this under
instruction from the rabbi.
All of this applies only to matters religious. Any person, Jew or gentile
may speak Israeli/Ivrit on a daily basis. Ivrit is a language like any
other. True, it has roots in biblical Hebrew. but it has none of the inherent
Holiness of biblical hebrew, Lashon Hakodesh. In fact Ivrit was established
to get Jews away from classical biblical Hebrew which was viewed as too
old-fashioned and, ironically, too "Jewish" (in the religious sense). So
really, Ashkenazim who use Israeli on a daily basis (for business ,
conversation, etc.) SHOULD use a different pronunciation for their prayers.
Those who are returning to Judaism and are not sure of their traditions and
mesorah should seek the advice of his/her rabbi.
(By the way, I have never heard the name of Abraham pronounced as Ovrohom.
The Aleph has a "Pasach" underneath which gives it the "Ah" sound, Perhaps,
Mr. Arons meant to type in "Avrohom" which is the normal Ashkenazi