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Yom Tov is slightly different than Shabbat. For example, on Shabbat one cannot cook, but one can cook on Yom Tov provided that it is from an existing flame.
My question is this... Let's say I have my computer or phone on, does this mean I could use the electrical equipment provided that it's not turned on/off?
It's no longer common practice to have halachic decisors permit the use of electricity. And while there may have been many decisors in the past that allowed the use of electricty, most of them are from 30+ years ago, describing things like the use of lightbulbs, microwaves, etc so i would not dare to claim how they might answer regarding a smart phone since they would have never seen one in their lifetime. But here is a list of every Rabbi that i'm aware of that allowed the use of electricity on Yom Tov, and you can go ahead and read their responsa to decide if any of them might apply to your question.
1903 Rabbi Yechiel Michel Epstein (the author of the Aruch Hashulchan) in Bet Va'ad LeHakhamim allows turning lights on on Yom Tob.
1903 Rabbi Yosef Yehoudah Strazberg (author of Yad Yosef, & Ab Bet Din of Makasov, Galitzia) in Bet Va'ad LeHakhamim also allows turning them on.
1912 Rabbi Refael Aharon Ben Shim'on (Chief Rabbi of Egypt) (He wrote this in 1901) in his UMitzor Debash allows turning them on.
1913 Rabbi Binyamin Aryeh HaKohen Weiss in his Eben Yeqarah allows turning them on.
1924 Rabbi Yehuda Yudil Rozenberg in his Maor HaHashmal in Montreal, Canada allows turning them on.
1932 Rabbi Ruben Margaliot in his Nefesh Hayah allows turning them on.
1934 Rabbi Yosef Messas (Rabbi of Tlemcen, Algeria and Meknes, Morocco and Haifa, Israel) in his Mayim Hayim allows turning them both on and off and he reiterated his position in numerous other places.
1934/35 Rabbi Tzvi Pesach Frank (Chief Rabbi of Jerusalem) in Qol Torah allows turning them on.
1935 Rabbi Ben Sion Meir Hai Uziel (The Rishon LeSion himself) in his Mishpete Uziel allows both turning them on and off and he reiterated this in 1947.
1936 Rabbi David HaKohen Saqli (Rosh Ab Bet Din in Oran, Algeria) in his Qiryat Hanah David (volume 2) allows both turning them on and off.
1945 Rabbi Eliezer Yehoudah Waldenberg in his famous Tzitz Eliezer (volume 1) allows turning them on.
1948 Rabbi Masoud HaKohen in his Pirhe Kehounah (Casablanca) allows turning them on.
1964 Rabbi Shraga Faivel Frank in his Toldot Ze-eb allows turning them on.
1973 Rabbi Shabetai Sheftel Weiss in his Hilkhita Rabeta LaShabeta allows turning them on. Source: List compiled by Joseph Mosseri.
The example normally given is an electric stove top which is already on before Yom Tov, can the stove top be changed in the same way that gas can be changed? The answer to that is that if the switch is a continuous rheostat, then it can be changed in the same way that gas can be changed. However, many stove tops have switches which actually "click" between settings. That is, you are, in effect, turning the electricity off and then on when the settings change. This is forbidden.
Similarly, turning it on is like lighting a new flame which is forbidden and not like lighting a gas flame from a pilot flame. That is why one must unplug the electronic ignition of a gas stove top before Yom Tov. One can only light the gas from a pre-existing flame.
The Halakhot of Stove and Oven Use on Shabbat and Yom Tov explains the modern "Sabbath Mode" ovens.
Mail Jewish has a related article which goes into detail on the subject.
As far as electricity goes, the only available leniency would be if you ruled with the minority opinion that it is treated as real fire. In that case, the Aruch Hashulchan, who did treat it as fire, has a responsa allowing one to shut off his light on yom tov sheini. But nobody I've heard of rules like that. In fact, his great grandson told me that his mother, the A.H.'s granddaughter did in fact act in accordance with that ruling allong with her family, but has since stopped after getting married. But getting back to your case, if you can establish that it is in fact fire and not let's say nolad or molad, you would then have to prove that the use of electricity is shava lichol nefesh and or rely upon mitoch. This would all be in order to actually manipulate the device. If in fact you would like to just leave the device on without touching it, you would still need to rule like everything mentioned just to get away from from any gezeira or harchaka so as not to come to manipulate it.
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On Yom Tov you can turn on and off the burners as needed for cooking if it was left on beforehand.
