Russia to Dublin, transiting Heathrow

The official information is rather confusing, but as far as I can see, you will be able to use your Irish visa to pass through UK immigration, provided the visa is endorsed with "BC" or "BC BIVS".

- ***Warning*** Passengers may make a landside transit if
holding a valid biometric visa issued by Ireland (Rep.)
endorsed "BC" or "BC BIVS" and traveling to Ireland (Rep.).
(SEE NOTE 57399)
NOTE 57399: E-visas or e-residence permits are not
accepted for landside transit.

Here is the complete information I just pulled off of TIMATIC, the exception for Irish visas is near the end.

Information as of  13MAY16 / 0958 UTC
National Russian Fed. (RU)      /Embarkation Russian Fed. (RU)
Transit United Kingdom (GB)
Destination Ireland (Rep.) (IE)
ALSO CHECK DESTINATION INFORMATION BELOW

United Kingdom (GB)

TWOV (Transit Without Visa):
Visa required, except for Nationals of Russian Fed. holding
confirmed onward tickets making an airside transit at London
Gatwick (LGW), London Heathrow (LHR) or Manchester (MAN) on
the same calendar day to a third country (excluding Ireland
(Rep.)).The following conditions must be complied with:
- passenger must arrive and depart by air; and
- passenger only intends to transit through the United
Kingdom; and
- passenger must hold all documents required for the next
destination; and
- passenger must not leave the transit area.
- OR:
Visa required, except for Passengers holding confirmed onward
tickets passing through United Kingdom immigration to make a
landside transit to a third country on a flight that departs
before 23:59 the next day. The following conditions must be
complied with:
- passenger must arrive and depart by air; and
- passenger must have no purpose in entering the United
Kingdom other than to pass through in transit; and
- passenger must hold all documents required for the next
destination; and
- passenger must pass through United Kingdom Immigration; and
- passenger travels with a document listed in the following
warning(s):
- ***Warning*** Passengers may make a landside transit if
holding a valid visa which allows entry into Australia,
Canada, New Zealand or USA and traveling as part of a
journey to, from or transit through the country that has
issued the visa.
Landside transit is not available if holding a US Visa Foil
type "YY" or "ZZ" endorsed "Not a visa. Foil prepared at
DHS request" nor the US Adit stamp worded "Processed for
I-551" (temporary form I-551).
- ***Warning*** Passengers may make a landside transit if
holding a valid common format "D" visa issued by an EEA
or Switzerland. (SEE NOTE 57399)
NOTE 57399: E-visas or e-residence permits are not
accepted for landside transit.
- ***Warning*** Passengers may make a landside transit if
holding a valid biometric visa issued by Ireland (Rep.)
endorsed "BC" or "BC BIVS" and traveling to Ireland (Rep.).
(SEE NOTE 57399)
NOTE 57399: E-visas or e-residence permits are not
accepted for landside transit.
Additional Information:
- Valid visas in expired travel documents are accepted if
accompanied by a new travel document.
- There is no passport control on traffic between Great
Britain and Northern Ireland and Ireland (Rep.).
- Leave to remain issued by Guernsey, Isle of Man or Jersey,
which are in the form of wet ink stamps, are also valid for
entry into or transit through the United Kingdom.

United Kingdom (GB)

Vaccinations not required.


Ireland (Rep.) (IE)

Passport required.
- Passports and other documents accepted for entry must be
valid for the period of intended stay.
Passport Exemptions:
- Passengers with an emergency or temporary passport.

Visa required, except for Nationals of Russian Fed. with a
valid "C" visa issued by the United Kingdom if they have first
entered the United Kingdom for a maximum stay of 90 days or
until the end of the period of stay granted in the United
Kingdom, whichever is shorter. (SEE NOTE 45576) (SEE NOTE
56994)
NOTE 45576: It is recommended that the passport contains
an "exempt vignette" issued by the United Kingdom and
obtained prior to travel.
NOTE 56994: Not applicable to "C" visas issued for the
purpose of transit, marriage or to enter into a civil
partnership.
Minors:
- A separate visa is required for children up to/incl. 16
- Visas issued to minors (persons under 18 years of age), with
a date of issue after 12 October 2014,
Additional Information:
- Visitors holding passports containing a British inadmissible
stamp could be refused entry.
- Visitors are required to hold proof of sufficient funds to
cover their stay.
Warning:
- Visitors not holding return/onward tickets could be refused
entry.

Ireland (Rep.) (IE)

Vaccinations not required.