There are numerous problems with bringing Korbanos today, even the Korban Pesach.
1) Tumah: Although a Korban Tzibur (and Pesach) can be sacrificed btumah, the Rambam paskens טומאה דחוייה בציבור and therefore בעי ריצוי ציץ. The Rambam also paskens דווקא בזמן שהוא על מצחו, which requires a Kohen Gadol, probably with all 8 begadim (or else he would be מחוסר בגדים) – which is an impossibility; we cannot make the Choshen nor the Avnet. (Another problem is that we are all מחוסר כפורים דזב, and טומאה הותרה בציבור may not override that)
2) Kohen Meyuchas: The Gemara explicitly forbids Kohanei Chazakah from performing the Avoidah in Kiddushin 79 and Kesuvos 24. The Chasam Sofer (Yo”d 236) brings the Ya’avetz (#91?) who says this is only a חומרא but not a reason to forgo the Avoida in a situation where there are no Kohanim Meyuchasim, but this is not at all a דבר מוסכם.
3) Mokom Mizbeach: The Gemara (Zevachim 62) writes that when rebuilding Bayis Sheni, they required a Navi to show them the Mokom Mizbeach. Whatever the reason they required a navi was, we don’t have one. (The Mahartz Chiyas has a Kuntrus Avoidas HaKodesh [located in Siman 74 of his Shaalos uTeshuvos] where he discusses this whole iyan, and he goes to lengths to differentiate between Bayis Sheni and today)
4) Bigdei Kehuna: Even if we rely on our Kohanim, the Bidgei Kehuna are מעכב. We do not have the required materials to create the Bigdei Kehuna. The Chasam Sofer writes that this is no problem, because although we are not able to make Bigdei Kohen Gadol because we lack tcheiles*, we can create Bidgei Kohen Hidyot. The problem is: (a) We need Bidgei Kohen Gadol too, because of Ritzui Tzitz (see Point 1), and (b) the Avnet of the Kohen Hidyot also requires Techeiles.
(* Even those who think we have techeiles will not be oiver on ספק כרת with that.)
5. ריח ניכוח: The Binyan Tzion (#1, 2), the Netziv (Ha’amek Davar Bechukosai) and the Chazon Ish (don’t remember exactly where; possible Zevachim 62?) explain the Pasuk ולא אריח בריח ניכוחכם to refer to Korbonos after the Churban. [This may not apply to Korbon Pesach)
6. The whole discussion only begins according to the Rambam, who holds that קדושה ראשונה קידשה לעתיד לבא. However, the Ra’avad disagrees, and holds that we cannot bring Korbonos today. This leads to many problem: (a) Possible wearing שעטנז [bigdei kahuna], (b) Possible entering Har HaBayis btumah [many poskim hold that even the Ra’avad agrees this is forbidden], (c) Possible damaging a Korbon שלא בשעת עבודה
These are big problems. R’ Chaim/Yechiel miPariz, Radvaz, The Chasam Sofer, Maharitz Chiyas, R’ Tzvi Hersch Kalisher didn’t seem to be bothered by these. There are possible ways out around these issue, although they require a lot of chiddushim.
Of course, this whole discussion is theoretical – even today, when different enemies control Har Habayis and we can gain access, it is guaranteed סכנה in terms of inviting retaliation, and that alone is reason not to bring the Korbon.
(And we don’t have anyone – at least who I am aware of – who has the expertise necessary to successfully bring the Karbanos. Practical issue.)